Problem 20 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}+8 x+16=7\)

Step-by-Step Solution

Answer

Solutions: \(x = -4 + \sqrt{7}\) and \(x = -4 - \sqrt{7}\).

Step 1: Set the equation to standard form

Start by writing the equation in the form where one side is a perfect square trinomial. The given equation is \(x^2 + 8x + 16 = 7\). To make one side a perfect square trinomial, we need to subtract 7 from both sides: \(x^2 + 8x + 16 - 7 = 0\). This simplifies to \(x^2 + 8x + 9 = 0\).

Step 2: Recognize the perfect square

Identify the perfect square on the left side of the equation. The expression \(x^2 + 8x + 16\) can be rewritten as \((x + 4)^2\). Thus, our equation becomes \((x + 4)^2 = 7\).

Step 3: Apply the Square Root Property

Apply the Square Root Property to solve \((x + 4)^2 = 7\). This property states that if \(a^2 = b\), then \(a = \pm\sqrt{b}\). Therefore, \(x + 4 = \pm\sqrt{7}\).

Step 4: Solve for x

Isolate \(x\) by subtracting 4 from both sides of the equation: \(x = -4 \pm \sqrt{7}\). This means the solutions are \(x = -4 + \sqrt{7}\) and \(x = -4 - \sqrt{7}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It can be written in the form \( (a + b)^2 \) or \( (a - b)^2 \). Here we expand these expressions as follows:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)
• \( (a - b)^2 = a^2 - 2ab + b^2 \)

In the given problem, our expression \( x^2 + 8x + 16 \) is a perfect square trinomial because it can be rewritten as \( (x + 4)^2 \). Here's why:

• The term \( x^2 \) suggests the use of \( x \) in our binomial.
• The term \( 8x \) tells us that \( 2a \times b \) is \( 8 \), therefore \( a = x \) and \( b = 4 \) (since \( 2 \times 4 = 8 \)).
• The term \( 16 \) is \( b^2 \), confirming \( b = 4 \) (since \( 4 \times 4 = 16 \)).

Recognizing perfect square trinomials helps simplify solving quadratic equations.

Solve Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. In this exercise, we were given the equation \(x^2+8x+16=7\). To solve it, follow these steps:

• First, rewrite the equation in such a way that it has a perfect square trinomial on one side, as done here to form \((x + 4)^2 = 7\).
• This simplification helps us easily apply further solving methods like the Square Root Method.

The target is to isolate \(x\) and find its possible values. Using proper techniques for rewriting and solving such equations is essential for finding the correct solutions.

Square Root Method

The Square Root Method is a powerful tool for solving quadratic equations that have been simplified to the form \( (x + a)^2 = b \). Here's how it works in the exercise we examined:

• Start with the equation \((x + 4)^2 = 7\).
• Apply the Square Root Property, which tells us if \(a^2 = b\), then \(a = \pm\sqrt{b}\).
• This process gives us \(x + 4 = \pm\sqrt{7}\).
• Isolate \(x\) by subtracting 4 from both sides: \(x = -4 \pm \sqrt{7}\).

This results in two solutions for \(x\): \(-4 + \sqrt{7}\) and \(-4 - \sqrt{7}\). This method is efficient and particularly useful when equations are structured to accommodate its application.