QuestionFactor the quadratic expression $x^{2} + 12x + 35$ .

Studdy Solution

STEP 1

1. The expression $x^2 + 12x + 35$ is a quadratic expression in standard form $ax^2 + bx + c$ .
2. The goal is to factor the expression into the form $(x + p)(x + q)$ where $p$ and $q$ are constants.
3. The sum of $p$ and $q$ should equal the coefficient of the linear term $b$ , and their product should equal the constant term $c$ .

STEP 2

1. Identify the coefficients.
2. Find two numbers that multiply to the constant term and add to the linear coefficient.
3. Write the factored form using these numbers.

STEP 3

Identify the coefficients from the quadratic expression $x^2 + 12x + 35$ .
- The coefficient of $x^2$ is $a = 1$ . - The coefficient of $x$ is $b = 12$ . - The constant term is $c = 35$ .

STEP 4

Find two numbers that multiply to $c = 35$ and add to $b = 12$ .
- Consider the pairs of factors of 35: $(1, 35)$ , $(5, 7)$ . - Check which pair adds to 12: $5 + 7 = 12$ .

STEP 5

Write the factored form using the numbers found in Step 2.
- Since $5$ and $7$ are the numbers that satisfy the conditions, the factored form is: $ (x + 5)(x + 7) \]
The factored form of the quadratic expression $x^2 + 12x + 35$ is:
$\boxed{(x + 5)(x + 7)}$

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