```Elementary Algebra Notes
Section 6.5
Page 1 of 3
Section 6.5: Solving Quadratic Equations by Factoring
Big Idea for section 6.5: Your skills in factoring can be applied to solving a new kind of nonlinear equation:
the quadratic equation, and this will require two new ways of thinking about how to solve equations:
1. We don’t try to isolate the variable right away.
2. You can get two different answers that are both solutions to the equation.
A quadratic equation in standard form is an equation where a polynomial of degree 2 is equal to zero.
Algebraic way of saying this: ax2 + bx + c = 0.
Note: the right hand side MUST be zero.
The Zero-Product Property
If the product of two numbers is zero, then at least one of the numbers is zero. That is,
if ab = 0, then a = 0 or b = 0, or both a and b are 0.
A consequence of the zero-product property is that some equations have more than one solution.
Practice:
1. Solve  x  4 3x  2  0
2. Solve x  2 x  4  0
Steps for Solving a Quadratic Equation:
1. Get the equation in standard form by expanding the polynomial equation (if needed), collecting all terms
on one side, and combining like terms.
3. Set each factor from step 2 equal to zero (this is justified by the zero-product property rule).
4. Solve each first degree equation for the variable.
5. Check answers in original equation.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 6.5
Page 2 of 3
Example: Solve the quadratic equation 2x2 – 5x + 7 = x2 + 1
1. Get the equation in standard form.
2 x2  5x  7  x2  1
2 x2  5x  7  x2  1  x2  1  x2  1
x2  5x  6  0
x2  5x  6  0
 x  2  x  3  0
3. Set each factor equal to zero.
 x  2  x  3  0
x  2  0 OR x  3  0
4. Solve each first degree equation.
x  2  0 OR x  3  0
x  2 OR x  3
5. Check answers in original equation.
2 x2  5x  7  x2  1
2  22  5  2  7  22  1
8  10  7  4  1
55
2 x2  5x  7  x2  1
2  32  5  3  7  32  1
18  15  7  9  1
10  10
Practice:
3. Solve x 2  2 x  8
4. Solve 2 x 2  x  6
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 6.5
Page 3 of 3
5. Solve 49 x 2  9  0
6. Solve x 2  3 x
7. Solve x  4 x  7   2
8. Solve  2 x  5 x  3  6 x
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
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