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.
2. Factor the quadratic polynomial.
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
2. Factor the quadratic polynomial.
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.