Transcript Document

5-3
Solving Quadratic Equations by
Graphing and Factoring
A trinomial (an expression with 3 terms) in standard
form (ax2 +bx + c) can be factored by finding factors
that multiply to equal c and add to equal b.
*Always check for a GCF first.
f(x) = x2 – 4x – 12
(x + 2)(x – 6)
Holt Algebra 2
c = -12, factors that
multiply to equal -12
b = -4, add the factors
and see which one
equals -4
1 and -12
1 + (-12) = -11
-1 and 12
-1 + 12 = 11
2 and -6
2 + (-6) = -4
-2 and 6
-2 + 6 = 4
3 and -4
3 + (-4) = -1
-3 and 4
-3 + 4 = 1
5-3
Solving Quadratic Equations by
Graphing and Factoring
Find the zeros of the function by factoring.
f(x)= x2 – 5x – 6
x2 – 5x – 6 = 0
(x + 1)(x – 6) = 0
x+1=0
x–6=0
x = –1 or x = 6
Holt Algebra 2
Set the function equal to 0.
Factor: Find factors of –6 that add to –5.
Set each parenthesis equal to zero
Solve each equation.
5-3
Solving Quadratic Equations by
Graphing and Factoring
Example 4B: Find Roots by Using Special Factors
Find the roots of the equation by factoring.
4x2 = 12x + 16
4x2 – 12x – 16 = 0
4(x2 – 3x – 4) = 0
4(x – 4)(x + 1) = 0
x–4=0
x=4
Holt Algebra 2
x+1=0
x = -1
Rewrite in standard form.
Factor. The GCF is 4.
Factor the trinomial
Set each parenthesis equal to zero
Solve each equation.
5-3
Solving Quadratic Equations by
Graphing and Factoring
Check It Out! Example 4a
Find the roots of the equation by factoring.
x2 – 4x = –4
x2 – 4x + 4 = 0
(x – 2)(x – 2) = 0
x–2=0
x=2
Holt Algebra 2
x–2=0
x=2
Rewrite in standard form.
Factor the perfect-square trinomial.
Set each parenthesis equal to zero.
Solve each equation.
5-3
Solving Quadratic Equations by
Graphing and Factoring
Write a quadratic function in standard form
with zeros 4 and –7.
x = 4 or x = –7
x – 4 = 0 or x + 7 = 0
(x – 4)(x + 7)
x2 + 3x – 28
f(x) = x2 + 3x – 28
Holt Algebra 2
Write the zeros as solutions for two
equations.
Rewrite each equation so that it
equals 0.
These two equations will represent the
parenthesis had you factored the function.
Multiply the binomials.
Name the function.
5-3
Solving Quadratic Equations by
Graphing and Factoring
Example 5 Continued
Check Graph the function
f(x) = x2 + 3x – 28
on a calculator. The
graph shows the
original zeros
of 4 and –7.
10
–10
10
–35
Holt Algebra 2
5-3
Solving Quadratic Equations by
Graphing and Factoring
Determine the equation of a quadratic
function with a double root at x = -1
x = -1
x+1=0
x = –1
x+1=0
(x + 1)(x + 1)
x2 +2x + 1
g(x) = x2 +2x + 1
Holt Algebra 2
Write the zeros as solutions for two
equations.
Rewrite each equation so that it
equals 0.
These two equations will represent the
parenthesis had you factored the function.
Multiply the binomials.
Name the function.