Question斉 Factor completely. $5 q^{2}+24 q-5$

Studdy Solution

STEP 1

1. The expression is a quadratic polynomial.
2. We are asked to factor the polynomial completely.
3. The polynomial can be factored into the product of two binomials.

STEP 2

1. Identify the form of the quadratic polynomial.
2. Use the method of factoring by grouping or trial and error to find the factors.
3. Verify the factors by expanding them back to the original polynomial.

STEP 3

Recognize that the polynomial $5q^2 + 24q - 5$ is in the standard quadratic form $ax^2 + bx + c$ , where $a = 5$ , $b = 24$ , and $c = -5$ .

STEP 4

To factor the quadratic, we need two numbers that multiply to $a \times c = 5 \times (-5) = -25$ and add to $b = 24$ .

STEP 5

The numbers that satisfy these conditions are $25$ and $-1$ because $25 \times (-1) = -25$ and $25 + (-1) = 24$ .

STEP 6

Rewrite the middle term $24q$ using the numbers found:
$5q^2 + 25q - q - 5$

STEP 7

Factor by grouping:
Group the terms:
$(5q^2 + 25q) + (-q - 5)$
Factor out the greatest common factor from each group:
$5q(q + 5) - 1(q + 5)$
Notice that $q + 5$ is a common factor:
$(5q - 1)(q + 5)$

STEP 8

Verify the factors by expanding:
$(5q - 1)(q + 5) = 5q^2 + 25q - q - 5 = 5q^2 + 24q - 5$
The factors are correct.
The completely factored form of the polynomial is:
$\boxed{(5q - 1)(q + 5)}$

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