Transcript Using the Zero-Product Property to Solve a Quadratic

Using the Zero-Product
Property to Solve a Quadratic
x–Intercepts, Solutions, Roots, and
Zeros in Quadratics
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.
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 )
x  14  0
x  14
6x 1  0
6 x  1
OR
x   16
Would you rather solve the equation above or this:
0 = x2 + 25x + 14 ?
Solving a Quadratic: Factoring
Solve: 0  2 x2  5x 12
Factor to
rewrite as
a product
c
4
8x
Product
(2x2)(-12)
-12
-24x2
ax2c
GCF ___
x
Use the ZeroProduct
Property
2x2
-3x
ax2
2x
-3
0   x  4 2x  3
x  4  0 2x  3  0
x  4
2x  3
3
x 2
8x
bx
-3x
5x
Sum
x  4 or
3
2
Solving a Quadratic: Already Factored
Solve: 6x  x  7  0
The equation is already factored AND it equals 0. Half
the work is already done. Just use the Zero-Product
Property
6x  x  7   0
x7  0
6x  0
x0
x7
x  0 or 7
Solving a Quadratic: Making Sure to
Isolate 0
Solve: 35  12 x 2  44 x
Factor to rewrite as a product
Product
c
5
Solve for 0 first!
35  12 x 2  44 x
0  12 x 2  44 x  35
30x
(2x2)(-12)
35
420x2
2
12x
GCF 2x
___
2
ax2c
14x
30x
ax
Use the Zero-Product Property
6x
0   2x  5 6x  7
2x  5  0 6x  7  0
2 x  5
6 x  7
x   52
x   76
7
bx
44x
Sum
x   or 
5
2
7
6
14x
Solving a Quadratic: Make Sure to
Isolate 0
Solve:  x  5 x  2  6
Factor to rewrite as a product
Product
c
Solve for 0 first!
 x  5 x  2  6
-4
-4x
-4
(x2)(-4)
-4x2
Distribute
x2  3x  10  6 GCF ___
x
x 2  3x  4  0
Use the Zero-Product Property
x2
ax2c
x
ax2
x
0   x  4 x  1
x 1  0
x40
x  1
x4
1
-4x
bx
-3x
Sum
x  4 or  1
1x
Do we Need Another Method?
Use the Zero Product Property to find the roots of:
x 2  3x  7  0
Product
c
-7
(x2)(-7)
But this
parabola has
two zeros.
-7x2
ax2c
___
x2
ax2
IMPOSSIBLE
bx
-3x
Sum
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.