Finding the Solutions, x-intercepts, Roots, or Zeros of

A Quadratic

x-intercept(s): Where the graph of y = ax2 + bx + c crosses the x-axis. The value(s) for x that makes a quadratic equal 0.

Solution(s) OR Roots: The value(s) of x that satisfies 0 = ax2 + bx + c.

Zeros: The value(s) of x that make ax2 + bx + c equal 0.

x–Intercepts, Solutions, Roots, and Zeros in Quadratics

Zero Product Property

If a . b = 0, then a and or b is equal to 0

Ex: Solve the following equation below.0 = ( x + 14 )( 6x + 1 )

Would you rather solve the equation above or this: 0 = 6x2 + 85x + 14 ?

14 0x

14x

6 1 0x

16x

6 1x

30x

14x

(2x2)(-12)

420x235

12x2ax2c

ax2

Example235 12 44x x

___ bx

44x

30x 14x

c Product

Sum

2x

6x 7

5

GCF

Solve:

2 5 0x 2 5x

6 7 0x 76x

6 7x

0 2 5 6 7x x 5 72 6 or x

Factor to rewrite as a product

Use the Zero-Product Property

20 12 44 35x x

235 12 44x x Solve for 0 first!

52x

(x2)(-7)

-7x2-7

x2

ax2c

ax2

Example

2 3 7 0x x

___ bx

-3x

IMPOSSIBLE

c

Product

Sum

Use the Zero Product Property to find the roots of:

But this parabola has

two zeros.

Just because a quadratic is not factorable, does not mean it does not have roots. Thus, there is a need for a new algebraic method

to find these roots.

Quadratic FormulaFor ANY 0 = ax2 + bx +c (standard form) the

value(s) of x is given by:

2 42

b b acxa

Opposite of b

“All Over”

MUST equal 0

Plus or Minus

This formula will provide the solutions (or lack thereof) to ANY Quadratic.

Example2 3 10 6x x Solve:

2 3 10 6x x Solve for 0 first!

2 3 4 0x x a = b = c = 1 -3 -4

Find the values of “a,” “b,” “c”

3 52

23 3 4 1 42 1x 3 25

2

Substitute into the Quadratic Formula

Simplify the expression in the square root first The square root

can be simplified.

3 52

3 52

Since the answers will be

rational, it is best to list both.

Or

82

22

41