College Algebra Tutorial 17


College Algebra
Tutorial 17: Quadratic Equations


WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Solve quadratic equations by factoring.
  2. Solve quadratic equations by the square root method.
  3. Solve quadratic equations by completing the square.
  4. Solve quadratic equations by using the quadratic formula.
  5. Find the discriminant of a quadratic equation and use it to tell how many and what type of solutions the equation has.




desk Introduction


 
In this tutorial, we will be looking at solving a specific type of equation called the quadratic equation.  The methods of solving these types of equations that we will take a look at are solving by factoring, by using the square root method, by completing the square, and by using the quadratic equation.  Sometimes one method won't work or another is just faster, depending on the quadratic equation given.  So it is good to know different ways to solve quadratic equations so you will be prepared for any type of situation.   After completing this tutorial, you will be a master at solving quadratic equations.  Solving equations in general is a very essential part of Algebra.  So I guess we better get to it.

 

 

desk Tutorial


  Quadratic Equation
Standard form: 

standard form
Where a does not equal 0.


 
Note that in Tutorial 14: Linear Equations in One Variable, we learned that a linear equation can be written in the form ax + b = 0 and that the exponent on the variable was 1.

Note that the difference is the highest exponent on the variable on the quadratic equation is 2. 

We are going to talk about four ways to solve quadratics.


 
  Solving Quadratic Equations by Factoring
 
You can solve a quadratic equation by factoring if, after writing it in standard form, the quadratic expression factors.

Step 1: Simplify each side if needed.
 

This would involve things like removing ( ), removing fractions, adding like terms, etc. 

To remove ( ):  Just use the distributive property.

To remove fractions: Since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

 

Step 2: Write in standard form, standard form, if needed.
 

If it is not in standard form, move any term(s) to the appropriate side by using the addition/subtraction property of equality. 

Also, make sure that the squared term is written first left to right, the x term is second and the constant is third and it is set equal to 0.

 

Step 3: Factor.
 

If you need a review on factoring go to Tutorial 7: Factoring Polynomials.

 

Step 4:  Use the Zero-Product Principle.
 

If ab = 0, then a = 0 or b = 0

0 is our magic number because the only way a product can become 0 is if at least one of its factors is 0. 

You can not guarantee what the factors would have to be if the product was set equal to any other number.  For example if ab = 1, then a = 5 and b = 1/5 or a = 3 and b = 1/3, etc.  But with the product set equal to 0, we can guarantee finding the solution by setting each factor equal to 0.  That is why it is important to get it in standard form to begin with. 

 

Step 5: Solve for the linear equation(s) set up in step 4. 
 

If a quadratic equation factors, it will factor into either one linear factor squared or two distinct linear factors.  So, the equations found in step 4 will be linear equations.  If you need a review on solving linear equations feel free to go to Tutorial 14: Linear Equations in One Variable.

 
 

notebookExample 1: Solve example 1a by factoring.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
This quadratic equation is already in standard form.

 
Step 3: Factor.

 
example 1b
*Quad. eq. in standard form
*Factor the trinomial

 
Step 4:  Use the Zero-Product Principle

AND

Step 5: Solve for the linear equation(s) set up in step 4. 


 
example 1c
*Use Zero-Product Principle
*Solve the first linear equation
 
 
 
 
 

*Solve the second linear equation
 

 


 
There are two solutions to this quadratic equation: x = -5 and x = 2.

 
 
 

notebookExample 2: Solve example 2a by factoring.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
example 2b
*Mult. both sides by LCD of 6 to clear fractions

 


 
Step 2: Write in standard form, standard form, if needed.

 
example 2c
*Inverse of add. 16 is sub. 16
*Quad. eq. in standard form

 
Step 3: Factor.

 
example 2d
*Quad. eq. in standard form
*Factor the diff. of two squares

 
Step 4:  Use the Zero-Product Principle

AND

Step 5: Solve for the linear equation(s) set up in step 4. 


 
example 2e
*Use Zero-Product Principle
*Solve the first linear equation
 
 
 
 
 
 
 
 
 
 
 

*Solve the second linear equation
 
 
 
 
 
 

 


 
There are two solutions to this quadratic equation: x = -4/5 and x = 4/5.

 
 
 

notebookExample 3: Solve example 3a by factoring.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
example 3b
*Use Dist. Prop. to clear the (  )

 
Step 2: Write in standard form, standard form, if needed.

 
example 3c

*Inverse of add. 2 is sub. 2
*Quad. eq. in standard form

 
Step 3: Factor.

 
example 3d
*Quad. eq. in standard form
*Factor the trinomial

 
Step 4:  Use the Zero-Product Principle

AND

Step 5: Solve for the linear equation(s) set up in step 4. 


 
example 3e
*Use Zero-Product Principle
*Solve the first linear equation
 
 
 
 
 
 
 
 
 
 
 

*Solve the second linear equation
 
 
 
 
 
 

 


 
There are two solutions to this quadratic equation: x = -2/3 and x = 1/2.

 
 
  Solving Quadratic Equations by the Square Root Method
 
You can solve a quadratic equation by the square root method if you can write it in the form square root method.

Step 1: Write the quadratic equation in the form square root methodif needed.
 

A and B represent algebraic expressions.  When you have the quadratic equation written in this form, it allows you to use the square root method described in step 2.

If it is not in this form, square root method, move any term(s) to the appropriate side by using the addition/subtraction or multiplication/division property of equality. 

 

Step 2: Apply the square root method.
 

If A and B are algebraic expressions such that square root method, then square root method,

also written square root method.
 

In other words, if you have an expression squared set equal to another expression, the inverse operation to solve it is to take the square root of both sides.  Since both a positive and its opposite squared result in the same answer, then you will have two answers, plus or minus the square root of B.

 

Step 3: Solve for the linear equation(s) set up in step 2. 
 

After applying the square root method to a quadratic equation you will end up with either one or two linear equations to solve.  Most times you will have two linear equations, but if  B is equal to 0, then you will only have one since plus or minus 0 is only one number.  If you need a review on solving linear equations feel free to go to Tutorial 14: Linear Equations in One Variable.

 
 

notebookExample 4: Solve example 4a by using the square root method.

videoView a video of this example


 
 
Step 1: Write the quadratic equation in the form square root methodif needed

AND

Step 2: Apply the square root method.


 
example 4b
*Written in the form square root method
*Apply the sq. root method
*There are 2 solutions

 
Step 3: Solve for the linear equation(s) set up in step 2. 

 
example 4c

*Sq. root of 16 = 4
 
 
 
 
 
 

*Neg. sq. root of 16 = - 4
 


 
There are two solutions to this quadratic equation: x = 4 and x = -4.

 
 
 

notebookExample 5: Solve example 5a by using the square root method.

videoView a video of this example


 
Step 1: Write the quadratic equation in the form square root method if needed

AND

Step 2: Apply the square root method.


 
Note how this quadratic equation is not in the form square root method to begin with.  The 5 is NOT part of the expression being squared on the left side of the equation.  We can easily write it in the form square root method by dividing both sides by 5.

 
example 5b
*Not in the form square root method
*Inv. of mult. by 5 is div. by 5

*Written in the form square root method

*Apply the sq. root method
*There are 2 solutions


 
Step 3: Solve for the linear equation(s) set up in step 2. 

 
example 5c
*Sq. root of 4 = 2
 
 
 
 
 

*Neg. sq. root of 4 = -2
 


 
There are two solutions to this quadratic equation: x = 2 and x = -2.

 
 
 

notebookExample 6: Solve example 6a by using the square root method.

videoView a video of this example


 
Step 1: Write the quadratic equation in the form square root method if needed

AND

Step 2: Apply the square root method.


 
example 6b
*Written in the form square root method

*Apply the sq. root method
*There are 2 solutions


 
Step 3: Solve for the linear equation(s) set up in step 2. 

 
example 6c

 

*Sq. root of 20 = 2 sq. root of 5
*Solve for x
 
 
 
 
 
 
 
 
 
 
 
 
 

*Neg. sq. root of 20 = -2 sq. root of 5
*Solve for x
 
 
 
 
 

 


 
There are two solutions to this quadratic equation: x example 6d  and x example 6e.

 
 
  Solving Quadratic Equations by Completing the Square
 
You can solve ANY quadratic equation by completing the square.  This comes in handy when a quadratic equation does not factor or is difficult to factor.


 

Step 1: Make sure that the coefficient on the x squared term is equal to 1.

If the coefficient of the x squared term is already 1, then proceed to step 2.

If the coefficient of the x squared term is not equal to 1, then divide both sides by that coefficient.


 

Step 2: Isolate the x squared and x terms.

In other words, rewrite it so that the x squaredand x terms are on one side and the constant is on the other  side.


 

Step 3: Complete the square.

At this point we will be creating a perfect square trinomial (PST).  Recall that a PST is a trinomial of the form perfect square trinomialand it factors in the form perfect square trinomial.  When it is in that form it will allow us to continue onto the next step and take the square root of both sides and find a solution.

We need to find a number that we can add to the x squaredand x terms so that we have a PST. 

We can get that magic number by doing the following:
 
 

If we have 
Complete the square

we can complete it’s square by adding the constant

complete the square

In other words, we complete the square by taking ½ of b (the coefficient of the x term) and then squaring it.  Make sure you remember to add it to BOTH sides to keep the equation balanced.


 

 

Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared.

If you need a review on factoring a perfect square trinomial, feel free to go to Tutorial 7: Factoring Polynomials.

 

Step 5: Solve the equation in step 4 by using the square root method.

 
 
 
 

notebookExample 7: Solve example 7a by completing the square.

videoView a video of this example


 
Step 1:Make sure that the coefficient on the x squared term is equal to 1.

 
The coefficient of the x squared term is already 1.

 
Step 2: Isolate the x squared and x terms.

 
The x squared and x terms are already isolated.

 
Step 3: Complete the square.

 
example 7b
*b is the coefficient of the x term

*Complete the square by taking 1/2 of b and squaring it
 

 


 
example 7c
*Add constant found above to BOTH sides of the eq.

*This creates a PST on the left side of eq. 


 
Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared.

 
example 7d

*Factor the PST

 
Step 5: Solve the equation in step 4 by using the square root method.

 
example 7e
*Written in the form square root method
*Apply the sq. root method
*There are 2 solutions
 
 
 
 
 
 
 
 
 
 
 

 


 
There are two solutions to this quadratic equation: x = 9  and x = 1.

 
 

notebookExample 8: Solve example 8a by completing the square.

videoView a video of this example


 
Step 1:Make sure that the coefficient on the x squared term is equal to 1.

 
Note how the coefficient on the x squared term is not 1 to begin with.  We can easily fix that by dividing both sides by that coefficient, which in this case is 3 .

 
example 8b

*Divide both sides by 3
 
 

*Coefficient of x squared term is now 1
 


 
Step 2: Isolate the x squared and x terms.

 
Note how the x squared and x terms are not isolated to begin with.  We can easily fix that by moving the constant to the other side of the equation.

 
example 8c
*Inverse of add. 3 is sub. 3

*x squared and x terms are now isolated
 


 
Step 3: Complete the square.

 
example 8d
*b is the coefficient of the x term
 
 

*Complete the square by taking 1/2 of b and squaring it
 
 
 
 

 


 
example 8e
*Add constant found above to BOTH sides of the eq.
 
 

*This creates a PST on the left side of eq. 
 


 
Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared.

 
example 8f

*Factor the PST

 
Step 5: Solve the equation in step 4 by using the square root method.

 
example 8g
*Written in the form square root method
*Apply the sq. root method
*There are 2 solutions
 

*Square root of a negative 1 is i
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

*Square root of a negative 1 is i
 
 
 
 
 

 


 
There are two solutions to this quadratic equation: x example 8h  and x example 8i.

 
  Solving Quadratic Equations by the Quadratic Formula

When standard form , then 

quadratic formula


 
You can solve ANY quadratic equation by using the quadratic formula.  This comes in handy when a quadratic equation does not factor or is difficult to factor.


 

Step 1: Simplify each side if needed.
 
This would involve things like removing ( ), removing fractions, adding like terms, etc. 

To remove ( ):  Just use the distributive property.

To remove fractions: Since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

 

Step 2: Write in standard form, standard form, if needed.
 

If it is not in standard form, move any term(s) to the appropriate side by using the addition/subtraction property of equality. 

Also, make sure that the squared term is written first left to right, the x term is second and the constant is third and it is set equal to 0.

 

Step 3: Identify a, b, and c.
 

When the quadratic equation is in standard form, standard form, then a is the coefficient in front of the xsquaredterm, b is the coefficient in front of the x term, and c is the constant term.

 

Step 4: Plug the values found in step 3 into the quadratic formula.
 

When standard form , then 

quadratic formula

 

Step 5: Simplify if possible. 


 
 
 

notebookExample 9: Solve example 9a by using the quadratic formula.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
This quadratic equation is already in standard form.

 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is 2.

b, the number in front of x, is -5.

c, the constant, is 1.
 

Make sure that you keep the sign that is in front of each of these numbers. 

Next we will plug it into the quadratic formula.   Note that we are only plugging in numbers, we don't also plug in the variable.


 
Step 4: Plug the values found in step 3 into the quadratic formula

AND

Step 5: Simplify if possible. 


 
example 9b
*Quadratic formula
 
 

*Plug in values found above for a, b, and c

*Simplify 
 
 

 


 
 
 
 

notebookExample 10: Solve example 10a by using the quadratic formula.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
This quadratic equation is already in standard form.

 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is 1.

b, the number in front of x, is 0. 
Note that b is 0 because the x term is missing.

c, the constant, is 9.
 

Make sure that you keep the sign that is in front of each of these numbers.

Next we will plug it into the quadratic formula.   Note that we are only plugging in numbers, we don't also plug in the variable.


 
Step 4: Plug the values found in step 3 into the quadratic formula

AND

Step 5: Simplify if possible. 


 
example 10b
*Quadratic formula
 
 

*Plug in values found above for a, b, and c

*Simplify 
 
 
 
 

*Square root of a negative 1 is i
 
 
 

 


 
 
 

notebookExample 11: Solve example 11a by using the quadratic formula.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
example 11b

*Inverse of add. 6x is sub. 6x

*Quad. eq. in standard form


 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is 1.

b, the number in front of x, is -6. 

c, the constant, is 9.
 

Make sure that you keep the sign that is in front of each of these numbers. 

Next we will plug it into the quadratic formula.   Note that we are only plugging in numbers, we don't also plug in the variable.


 
Step 4: Plug the values found in step 3 into the quadratic formula

AND

Step 5: Simplify if possible. 


 
example 11c
*Quadratic formula
 
 
 

*Plug in values found above for a, b, and c

*Simplify 
 
 
 
 
 
 
 

 


 
 
  Discriminant
 
When a quadratic equation is in standard form, standard form, the expression, discriminant, that is found under the square root part of the quadratic formula is called the discriminant. 

The discriminant can tell you how many solutions there are going to be and if the solutions are real numbers or complex imaginary numbers.


  Discriminant
discriminant Kinds of solution for
standard form discriminant Two distinct real solutions

Note that the value of the discriminant is found under the square root and there is a + or - in front of it.  So, if that value is positive, then there would be two distinct  real number answers.

discriminant One real solution

Note that the value of the discriminant is found under the square root and there is a + or - in front of it.  So, if that value is zero, + or - zero is the same number, so there would be only one real number solution.

discriminant Two distinct complex imaginary solution

Note that the value of the discriminant is found under the square root and there is a + or - in front of it.  So, if that value is negative, then there would be two distinct complex imaginary number answers.


 
  Finding the Discriminant
 
Step 1: Simplify each side if needed.
 
This would involve things like removing ( ), removing fractions, adding like terms, etc. 

To remove ( ):  Just use the distributive property.

To remove fractions: Since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

 

Step 2: Write in standard form, standard form, if needed.
 

If it is not in standard form, move any term(s) to the appropriate side by using the addition/subtraction property of equality.

Also, make sure that the squared term is written first left to right, the x term is second and the constant is third and it is set equal to 0.

 

Step 3: Identify a, b, and c.
 

When the quadratic equation is in standard form, standard form, then a is the coefficient in front of the x squaredterm, b is the coefficient in front of the x term, and c is the constant term.

 

Step 4: Plug the values found in step 3 into the discriminant, discriminant.
 

 

Step 5: Simplify if possible. 


 
 
 

notebookExample 12: Find the discriminant.  Based on the discriminant, indicate how many and what type of solutions there would be. example 12a

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
This quadratic equation is already in standard form.

 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is 3.

b, the number in front of x, is 1.

c, the constant, is 10.
 

Make sure that you keep the sign that is in front of each of these numbers. 


 
Step 4: Plug the values found in step 3 into the discriminant, discriminant.

AND

Step 5: Simplify if possible. 


 
example 12b
*Discriminant formula

*Plug in values found above for a, b, and c

*Discriminant


 
Since the discriminant is a negative number, that means there are two distinct complex imaginary solutions.

 
 
 

notebookExample 13: Find the discriminant.  Based on the discriminant, indicate how many and what type of solutions there would be. example 13a.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
example 13b

*Inverse of sub. 16 is add. 16

*Quad. eq. in standard form


 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is 1.

b, the number in front of x, is -8.

c, the constant, is 16.
 

Make sure that you keep the sign that is in front of each of these numbers. 


 
Step 4: Plug the values found in step 3 into the discriminant, discriminant.

AND

Step 5: Simplify if possible. 


 
example 13c
*Discriminant formula

*Plug in values found above for a, b, and c

*Discriminant


 
Since the discriminant is zero, that means there is only one real number solution.

 
 
 

notebookExample 14: Find the discriminant.  Based on the discriminant, indicate how many and what type of solutions there would be. example 14a.

videoView a video of this example


 
Step 1: Simplify each side if needed.

 
This quadratic equation is already simplified.

 
Step 2: Write in standard form, standard form, if needed.

 
example 14b

*Inverse of sub. 7x is add. 7x

*Quad. eq. in standard form


 
Step 3: Identify a, b, and c.

 
a, the number in front of x squared, is -5.

b, the number in front of x, is 7.

c, the constant, is 0.
Note that since the constant is missing it is understood to be 0.
 

Make sure that you keep the sign that is in front of each of these numbers. 


 
Step 4: Plug the values found in step 3 into the discriminant, discriminant.

AND

Step 5: Simplify if possible. 


 
example 14c
*Discriminant formula

*Plug in values found above for a, b, and c

*Discriminant


 
Since the discriminant is a positive number, that means there are two distinct real number solutions.

 

 

desk Practice Problems


 
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1b: Solve by factoring.
 

1a. Problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

 

 

pencil Practice Problems 2a - 2b: Solve by using the square root method.

 
2a. problem 2a
(answer/discussion to 2a)
2b. problem 2b
(answer/discussion to 2b)

 

 

pencil Practice Problems 3a - 3b: Solve by completing the square.
 

3a. problem 3a
(answer/discussion to 3a)
3b. problem 3b
(answer/discussion to 3b)

 

 

pencil Practice Problems 4a - 4c: Solve by using the quadratic equation.
 

4a. problem 4a
(answer/discussion to 4a)
4b. problem 4b
(answer/discussion to 4b)

 
4c. problem 4c
(answer/discussion to 4c)

 

 

pencil Practice Problems 5a - 5c: Find the discriminant. Based on the discriminate, indicate how many and what type of solutions there would be.
 

5a. problem 5a
(answer/discussion to 5a)
5b. problem 5b
(answer/discussion to 5b)

 
5c. problem 5c
(answer/discussion to 5c)

 

 

desk Need Extra Help on these Topics?



 
The following are webpages that can assist you in the topics that were covered on this page:
 

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut30_eqfact.htm
This webpage helps you with solving polynomial equations by factoring.

http://www.purplemath.com/modules/solvquad.htm
This webpage helps with solving quadratic equations by factoring

http://www.sosmath.com/algebra/quadraticeq/sobyfactor/ sobyfactor.html 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut99.htm 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut105.htm 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut110.htm 
This webpage helps with solving quadratic equations by factoring. 

http://www.purplemath.com/modules/solvquad2.htm
This webpage helps with solving quadratic equations by taking the square root.

http://www.purplemath.com/modules/solvquad3.htm
This webpage helps with solving quadratic equations by completing the square.

http://www.purplemath.com/modules/solvquad4.htm
This webpage helps with solving quadratic equations by using the quadratic formula.


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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