Learning Objectives
Introduction
Tutorial
,
Where a does not equal 0.
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.
Step 1: Simplify each
side if needed.
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, ,
if needed.
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.
Step 4:
Use the
Zero-Product Principle.
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.
Example 1: Solve by factoring.
AND
Step 5: Solve for the linear equation(s) set up in step 4.
*Solve the second linear
equation
Example 2: Solve by factoring.
AND
Step 5: Solve for the linear equation(s) set up in step 4.
*Solve the second linear
equation
Example 3: Solve by factoring.
AND
Step 5: Solve for the linear equation(s) set up in step 4.
*Solve the second linear
equation
Step 1: Write the
quadratic equation in the form if
needed.
If it is not in this form, , 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.
also written .
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.
Example 4: Solve by using the square root method.
AND
Step 2: Apply the square root method.
*Neg. sq. root of 16 = - 4
Example 5: Solve by using the square root method.
AND
Step 2: Apply the square root method.
*Written in the form
*Apply the sq. root method
*There are 2 solutions
*Neg. sq. root of 4 = -2
Example 6: Solve by using the square root method.
AND
Step 2: Apply the square root method.
*Apply the sq. root method
*There are 2 solutions
*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
If the coefficient of the term is not equal to 1, then divide both sides by that coefficient.
We need to find a number that we can add to the and x terms so that we have a PST.
We can get that magic number by
doing the following:
we can complete it’s square by adding the constant
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.
Example 7: Solve by completing the square.
*Complete the square by taking
1/2 of b and squaring it
*This creates a PST on the left side of eq.
Example 8: Solve by completing the square.
*Coefficient of term is now 1
* and x terms are now isolated
*Complete the square by taking
1/2 of b and squaring it
*This creates a PST on the left
side of eq.
*Square
root of a negative 1 is i
*Square
root of a negative 1 is i
When , then
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, ,
if needed.
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.
Step 4: Plug the
values found in step 3 into the quadratic formula.
Step 5: Simplify if possible.
Example 9: Solve by using the quadratic formula.
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Simplify
Example 10: Solve by using the quadratic formula.
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Simplify
*Square
root of a negative 1 is i
Example 11: Solve by using the quadratic formula.
*Quad. eq. in standard form
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Simplify
The discriminant can tell you how many solutions there are going to be and if the solutions are real numbers or complex imaginary numbers.
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.
One real solutionNote 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.
Two distinct complex imaginary solutionNote 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.
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, ,
if needed.
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.
Step 4: Plug the
values found in step 3 into the discriminant, .
Step 5: Simplify if possible.
Example 12: Find the discriminant. Based on the discriminant, indicate how many and what type of solutions there would be.
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Discriminant
Example 13: Find the discriminant. Based on the discriminant, indicate how many and what type of solutions there would be. .
*Quad. eq. in standard form
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Discriminant
Example 14: Find the discriminant. Based on the discriminant, indicate how many and what type of solutions there would be. .
*Quad. eq. in standard form
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.
AND
Step 5: Simplify if possible.
*Plug in values found above for a, b, and c
*Discriminant
Practice Problems
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.
Practice Problems 1a - 1b: Solve by factoring.
Practice Problems 2a - 2b: Solve by using the square root method.
Practice Problems 3a - 3b: Solve by completing the square.
Practice Problems 4a - 4c: Solve by using the quadratic equation.
Practice Problems 5a - 5c: Find the discriminant. Based on the discriminate, indicate how many and what type of solutions there would be.
Need Extra Help on these Topics?
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.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 17, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward.
All rights reserved.