QuestionFill in the gaps to factorise this expression. $x^{2}+6 x-27=(x-\not)(x+\ldots)$

Studdy Solution

STEP 1

1. We are given a quadratic expression $x^2 + 6x - 27$ to factor.
2. The expression can be factored into the form $(x - a)(x + b)$ .
3. We need to find the values of $a$ and $b$ such that the expression is equivalent to the original quadratic.

STEP 2

1. Identify the values of $a$ and $b$ that satisfy the conditions for factoring a quadratic expression.
2. Verify the factorization by expanding the factored form to ensure it matches the original expression.

STEP 3

Identify the values of $a$ and $b$ such that:
$a \cdot b = -27$ $-a + b = 6$
We need two numbers whose product is $-27$ and whose sum is $6$ .

STEP 4

List the factor pairs of $-27$ and find the pair that sums to $6$ :
Possible pairs: $(-1, 27), (1, -27), (-3, 9), (3, -9)$ .
The pair $(-3, 9)$ satisfies the condition $-3 + 9 = 6$ .

STEP 5

Write the factorization using the identified values:
$x^2 + 6x - 27 = (x - (-3))(x + 9)$
Simplify the expression:
$(x + 3)(x + 9)$

STEP 6

Verify the factorization by expanding:
$(x + 3)(x + 9) = x^2 + 9x + 3x + 27 = x^2 + 12x + 27$
The factorization is incorrect. Re-evaluate the factor pairs.

STEP 7

Re-evaluate the factor pairs of $-27$ and find the correct pair:
The correct pair should be $(3, -9)$ because:
$3 \cdot (-9) = -27$ $3 + (-9) = -6$
The signs were incorrect in the previous step.

STEP 8

Write the correct factorization:
$x^2 + 6x - 27 = (x - 3)(x + 9)$
Verify by expanding:
$(x - 3)(x + 9) = x^2 + 9x - 3x - 27 = x^2 + 6x - 27$
The factorization is now correct.

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