Question

match each equation with its factored form. then match each factored form with its solution.

equation	factored form	solutions
$x^2 + 10x + 18 = -3$	$\boxed{\quad}$	$\boxed{\quad}$

$\boldsymbol{\because (x + 4)(x + 6) = 0}$ $\boldsymbol{\because (x + 3)(x + 7) = 0}$ $\boldsymbol{\because x = -4 \text{ and } x = -6}$ $\boldsymbol{\because x = -3 \text{ and } x = -7}$

Explanation:

Step1: Solve \(x^2 + 10x + 3 = -21\)

First, rewrite the equation in standard form: \(x^2 + 10x + 3 + 21 = 0\)
\(x^2 + 10x + 24 = 0\)
Factor the quadratic: We need two numbers that multiply to 24 and add to 10. Those numbers are 4 and 6.
So, \((x + 4)(x + 6) = 0\)
Set each factor equal to zero: \(x + 4 = 0\) or \(x + 6 = 0\)
Solutions: \(x = -4\) and \(x = -6\)

Step2: Solve \(x^2 + 10x + 18 = -3\)

Rewrite in standard form: \(x^2 + 10x + 18 + 3 = 0\)
\(x^2 + 10x + 21 = 0\)
Factor the quadratic: We need two numbers that multiply to 21 and add to 10. Those numbers are 3 and 7.
So, \((x + 3)(x + 7) = 0\)
Set each factor equal to zero: \(x + 3 = 0\) or \(x + 7 = 0\)
Solutions: \(x = -3\) and \(x = -7\)

Answer:

For the equation \(x^2 + 10x + 3 = -21\):

• Factored Form: \((x + 4)(x + 6) = 0\)
• Solutions: \(x = -4\) and \(x = -6\)

For the equation \(x^2 + 10x + 18 = -3\):

• Factored Form: \((x + 3)(x + 7) = 0\)
• Solutions: \(x = -3\) and \(x = -7\)