Problem 11 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ x^{2}+6 x+9=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2}+6x+9=0\) is \(x=-3\).

Step 1: Identify The Factors

Look for two numbers that multiply to give 9 (the constant term) and add up to 6 (the coefficient of \(x\)). Here, both numbers are 3, since \(3 \times 3 = 9\) and \(3 + 3 = 6\).

Step 2: Factor The Quadratic

Rewrite the quadratic equation \(x^{2}+6 x+9\) by expressing the middle term \((6x)\) as the sum of \(3x\) and \(3x\). This gives \(x^{2} + 3x + 3x + 9\). Now group the first two terms together and the last two terms together to create \(x (x + 3) + 3(x + 3)\), which is the factored form \((x + 3)^2\).

Step 3: Solve The Factored Equation

Set each factor equal to zero and solve for \(x\). In this case, the factored equation \((x + 3)^2 = 0\) gives only one solution. So, solving \(x + 3 = 0\) yields \(x = -3\).