Problem 26 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Quadratic Equation

Note that the equation \( x^{2}+6x+9=0 \) is a quadratic equation, because the highest power of the variable (x) is 2. The coefficients for this equation are \(a = 1\), \(b = 6\), and \(c = 9\).

Step 2: Factor The Equation

Now factor the quadratic equation in the form \( ax^2 + bx + c = 0 \). The equation \( x^{2}+6x+9=0 \) can be factored into \( (x+3)^2 = 0 \). This is often done by inspection, or by using a method such as 'completing the square'.

Step 3: Solve For The Variable

Having factored the equation, the next step is to use this mere fact that if the product of two factors equals zero, then at least one of the factors must be zero. So, set the factor equal to zero and solve for the variable: \( x+3 = 0 \). Therefore, \(x = -3\).

Key Concepts

Factoring Quadratic Equations

When you encounter a quadratic equation like
\( x^{2} + 6x + 9 = 0 \), the method of factoring is a straightforward approach to find the variable's values that satisfy the equation. The idea is to express the quadratic expression as a product of two binomials. To factor effectively, understanding patterns, like the difference of squares and perfect square trinomials, is essential.

For the given equation, it has the form of a perfect square trinomial because it satisfies the pattern \( a^2 + 2ab + b^2 = (a + b)^2 \). Here, \( a = x \) and \( b = 3 \), leading to the factored form \( (x + 3)^2 \). Recognizing such patterns can make the factoring process more efficient.

Quadratic Formula

In cases where factoring might be challenging, the quadratic formula is a powerful tool that can solve any quadratic equation. The formula is derived from the general form of a quadratic equation, \( ax^2 + bx + c = 0 \), and is stated as
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

This formula computes the roots by considering the coefficients of the equation. The discriminant \( b^2 - 4ac \) inside the square root determines the nature of the roots. If it’s positive, there are two real and distinct solutions. If it equals zero, there is one real, repeated solution. If it's negative, the equation has two complex solutions. In the case of our equation \( x^{2} + 6x + 9 = 0 \), the discriminant would be zero, confirming that the solution is a perfect square and thus a repeated real root.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial plus a constant. To complete the square for an equation like \( x^{2} + 6x + 9 = 0 \), you would:

• Ensure the coefficient of \( x^{2} \) is 1 (if not, divide the whole equation by that coefficient first).
• Move the constant term to the other side of the equation.
• Add \( (b/2)^2 \) to both sides of the equation to form \( (x + b/2)^2 \) on one side.

For our equation, \( (6/2)^2 = 9 \) is already on the right side, demonstrating that the equation is already a complete square form.

Zero Product Property

The zero product property states that if a product of two factors equals zero, then at least one of the factors must be zero. This property is fundamental when working with factored forms of quadratic equations, like \( (x+3)^2 = 0 \).

It allows for setting each factor equal to zero and solving for the variable. Using our example, we only have one factor, but if we had two, such as \( (x+3)(x-2) = 0 \), we would set \( x+3 = 0 \) and \( x-2 = 0 \) and solve for \( x \) to find the two roots of the equation. Since \( (x + 3)^2 = 0 \), we only need to solve the equation \( x + 3 = 0 \) which gives the single solution \( x = -3 \).