Question2. Write in factored form. a) $f(x)=x^{2}-81$ b) $f(x)=6 x^{2}+5 x-4$

Studdy Solution

STEP 1

1. We are asked to write each given quadratic expression in its factored form.
2. The expressions can be factored using algebraic identities and methods like the difference of squares and factoring trinomials.

STEP 2

1. Factor the expression $f(x) = x^2 - 81$ using the difference of squares.
2. Factor the expression $f(x) = 6x^2 + 5x - 4$ using the method of factoring trinomials.

STEP 3

Recognize that $f(x) = x^2 - 81$ is a difference of squares. The difference of squares formula is $a^2 - b^2 = (a - b)(a + b)$ .
$x^2 - 81 = (x - 9)(x + 9)$

STEP 4

For $f(x) = 6x^2 + 5x - 4$ , use the method of factoring trinomials. Look for two numbers that multiply to $6 \times (-4) = -24$ and add to $5$ .
The numbers are $8$ and $-3$ .

STEP 5

Rewrite the middle term $5x$ using the numbers found:
$6x^2 + 8x - 3x - 4$

STEP 6

Group the terms and factor by grouping:
$(6x^2 + 8x) + (-3x - 4)$
Factor out the greatest common factor from each group:
$2x(3x + 4) - 1(3x + 4)$

STEP 7

Notice that $(3x + 4)$ is a common factor:
$(2x - 1)(3x + 4)$
The factored forms are: a) $f(x) = (x - 9)(x + 9)$ b) $f(x) = (2x - 1)(3x + 4)$

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