Answer :
To factor the expression [tex]\(25x^2 - 90x - 40\)[/tex] completely, follow these steps:
1. Identify a Common Factor: First, check if there is a greatest common factor (GCF) for the entire expression. Here, all the coefficients (25, -90, and -40) are divisible by 5. So, factor out 5:
[tex]\[
25x^2 - 90x - 40 = 5(5x^2 - 18x - 8)
\][/tex]
2. Factor the Quadratic Expression: Now, focus on factoring the quadratic expression [tex]\(5x^2 - 18x - 8\)[/tex]. We'll look for two numbers that multiply to [tex]\(5 \times (-8) = -40\)[/tex] and add to [tex]\(-18\)[/tex].
3. Find the Numbers: Consider the products that equal -40:
- Possible pairs: (1, -40), (2, -20), (4, -10), (-1, 40), (-2, 20), (-4, 10)
- The pair that adds to -18 is (-20, 2).
4. Rewrite the Middle Term: Use the numbers -20 and 2 to break down the middle term of the quadratic expression:
[tex]\[
5x^2 - 18x - 8 = 5x^2 - 20x + 2x - 8
\][/tex]
5. Factor by Grouping: Now, group the terms and factor them:
[tex]\[
= (5x^2 - 20x) + (2x - 8)
\][/tex]
Factor out the greatest common factor from each group:
[tex]\[
= 5x(x - 4) + 2(x - 4)
\][/tex]
6. Factor the Common Binomial: Notice that [tex]\(x - 4\)[/tex] is a common factor in both groups:
[tex]\[
= (5x + 2)(x - 4)
\][/tex]
7. Combine: Thus, the completely factored form of the original expression is:
[tex]\[
5(5x + 2)(x - 4)
\][/tex]
So, the expression [tex]\(25x^2 - 90x - 40\)[/tex] factors completely to [tex]\(5(x - 4)(5x + 2)\)[/tex].
1. Identify a Common Factor: First, check if there is a greatest common factor (GCF) for the entire expression. Here, all the coefficients (25, -90, and -40) are divisible by 5. So, factor out 5:
[tex]\[
25x^2 - 90x - 40 = 5(5x^2 - 18x - 8)
\][/tex]
2. Factor the Quadratic Expression: Now, focus on factoring the quadratic expression [tex]\(5x^2 - 18x - 8\)[/tex]. We'll look for two numbers that multiply to [tex]\(5 \times (-8) = -40\)[/tex] and add to [tex]\(-18\)[/tex].
3. Find the Numbers: Consider the products that equal -40:
- Possible pairs: (1, -40), (2, -20), (4, -10), (-1, 40), (-2, 20), (-4, 10)
- The pair that adds to -18 is (-20, 2).
4. Rewrite the Middle Term: Use the numbers -20 and 2 to break down the middle term of the quadratic expression:
[tex]\[
5x^2 - 18x - 8 = 5x^2 - 20x + 2x - 8
\][/tex]
5. Factor by Grouping: Now, group the terms and factor them:
[tex]\[
= (5x^2 - 20x) + (2x - 8)
\][/tex]
Factor out the greatest common factor from each group:
[tex]\[
= 5x(x - 4) + 2(x - 4)
\][/tex]
6. Factor the Common Binomial: Notice that [tex]\(x - 4\)[/tex] is a common factor in both groups:
[tex]\[
= (5x + 2)(x - 4)
\][/tex]
7. Combine: Thus, the completely factored form of the original expression is:
[tex]\[
5(5x + 2)(x - 4)
\][/tex]
So, the expression [tex]\(25x^2 - 90x - 40\)[/tex] factors completely to [tex]\(5(x - 4)(5x + 2)\)[/tex].
