1. Quadratic Equations P.7

2. Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written in the standard form
ax2 + bx + c = 0
where a, b, and c are real numbers with a not equal to 0. A quadratic equation in x is also called a second-degree polynomial equation in x.

3. The Zero-Product Principle
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.

4. Solving a Quadratic Equation by Factoring
If necessary, rewrite the equation in the form ax2 + bx + c = 0, moving all terms to one side, thereby obtaining zero on the other side.
Factor.
Apply the zero-product principle, setting each factor equal to zero.
Solve the equations in step 3.
Check the solutions in the original equation.

5. Text Example
Solve 2x2 + 7x = 4 by factoring and then using the zero-product principle.
Step Move all terms to one side and obtain zero on the other side. Subtract 4 from both sides and write the equation in standard form.
2x2 + 7x - 4 = 4 - 4
2x2 + 7x - 4 = 0
Step Factor.
(2x - 1)(x + 4) = 0

6. Solution cont.
Solve 2x2 + 7x = 4 by factoring and then using the zero-product principle.
Steps 3 and Set each factor equal to zero and solve each resulting equation.
2 x - 1 = 0 or x + 4 = 0
2 x = 1 x = -4
x = 1/2
Steps 5 check your solution

7. Example (2x + -3)(2x + 1) = 5 4x2 - 4x - 3 = 5 4x2 - 4x - 8 = 0
x - 2 = 0, and x + 1 = 0
So x = 2, or -1

8. The Square Root Method
If u is an algebraic expression and d is a positive real number, then u2 = d has exactly two solutions.
If u2 = d, then u = d or u = -d
Equivalently,
If u2 = d then u = d

9. Completing the Square
If x2 + bx is a binomial then by adding (b/2) 2, which is the square of half the coefficient of x, a perfect square trinomial will result. That is,
x2 + bx + (b/2)2 = (x + b/2)2

10. Text Example
What term should be added to the binomial x2 + 8x so that it becomes a perfect square trinomial? Then write and factor the trinomial.
The term that should be added is the square of half the coefficient of x. The coefficient of x is 8. Thus, (8/2)2 = 42. A perfect square trinomial is the result.
x2 + 8x + 42 = x2 + 8x + 16 = (x + 4)2

11. Don’t forget to add to both sides
B divided by 2, squared
Don’t forget to add to both sides

12. Quadratic Formula

15. The Discriminant and the Kinds of Solutions to ax2 + bx +c = 0
No x-intercepts
No real solution;
two complex imaginary solutions
b2 – 4ac < 0
One x-intercept
One real solution
(a repeated solution)
b2 – 4ac = 0
Two x-intercepts
Two unequal real solutions
b2 – 4ac > 0
Graph of
y = ax2 + bx + c
Kinds of solutions
to ax2 + bx + c = 0
Discriminant
b2 – 4ac