To factor the quadratic expression [tex]\( 15x^2 - 4x - 4 \)[/tex] completely, let's go through the steps:
1. Identify the coefficients: For the quadratic expression [tex]\( 15x^2 - 4x - 4 \)[/tex], the coefficients are:
- [tex]\( a = 15 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = -4 \)[/tex]
2. Multiply [tex]\( a \)[/tex] and [tex]\( c \)[/tex]: Calculate [tex]\( ac \)[/tex]:
[tex]\[
ac = 15 \times -4 = -60
\][/tex]
3. Find two numbers that multiply to [tex]\( ac \)[/tex] and add to [tex]\( b \)[/tex]: We need two numbers that multiply to [tex]\(-60\)[/tex] and add up to [tex]\(-4\)[/tex]. Those numbers are [tex]\( -10 \)[/tex] and [tex]\( 6 \)[/tex], because:
[tex]\[
-10 \times 6 = -60
\][/tex]
[tex]\[
-10 + 6 = -4
\][/tex]
4. Rewrite the middle term using these two numbers: Rewrite the quadratic expression [tex]\( 15x^2 - 4x - 4 \)[/tex] by splitting the middle term using [tex]\(-10x\)[/tex] and [tex]\(6x\)[/tex]:
[tex]\[
15x^2 - 10x + 6x - 4
\][/tex]
5. Group the terms: Group the terms in pairs:
[tex]\[
(15x^2 - 10x) + (6x - 4)
\][/tex]
6. Factor out the greatest common factor (GCF) from each group:
- From [tex]\( 15x^2 - 10x \)[/tex], the GCF is [tex]\( 5x \)[/tex]:
[tex]\[
5x(3x - 2)
\][/tex]
- From [tex]\( 6x - 4 \)[/tex], the GCF is [tex]\( 2 \)[/tex]:
[tex]\[
2(3x - 2)
\][/tex]
7. Factor out the common binomial factor: Notice that both terms have a common binomial factor [tex]\( 3x - 2 \)[/tex]:
[tex]\[
5x(3x - 2) + 2(3x - 2) = (5x + 2)(3x - 2)
\][/tex]
8. Write the final factored expression: Thus, the completely factored form of the quadratic expression [tex]\( 15x^2 - 4x - 4 \)[/tex] is:
[tex]\[
(3x - 2)(5x + 2)
\][/tex]
So, the final answer is:
[tex]\[
15x^2 - 4x - 4 = (3x - 2)(5x + 2)
\][/tex]
