Problem 69 - Elementary Algebra

Question

Solve by completing the square. \(v^{2}+6 v=40\)

Step-by-Step Solution

Answer

The solutions are \(v=4\) and \(v=-10\).

Step 1: Move Constant Term

First, move the constant term to the right side of the equation. This will isolate the quadratic and linear terms on the left side.Given equation: i.e., \[v^{2}+6v=40\]

Step 2: Form a Perfect Square

To complete the square, take the coefficient of the linear term (6), divide by 2, and then square it. Add and subtract this square to/from the left side of the equation.Half of 6 is 3, and 3 squared is 9.So, add and subtract 9:\[v^{2} + 6v + 9 - 9 = 40\]\[v^{2}+6v+9=49\]

Step 3: Simplify the Left Side

Rewrite the left side as a square of a binomial. The expression \(v^{2} + 6v + 9\) is the same as \((v + 3)^2\).So, the equation becomes:\[(v + 3)^{2} = 49\]

Step 4: Solve for the Variable

Take the square root of both sides of the equation:\[\begin{cases} v+3=7 \ v+3=-7 \ \end{cases}\]Now solve each equation for \(v\):1) \(v + 3 = 7\)➡\(v = 4\)2) \(v + 3 = -7\)➡\(v = -10\)

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form a x ^ { 2 } + b x + c = 0 , where x is the variable, and a, b , and c are constants. These equations are called quadratic because the highest power of the variable x is 2. Solving quadratic equations is a fundamental skill in algebra. Understanding how to work with these equations helps us solve many real-world problems. Some common techniques for solving quadratic equations include:

• Factoring
• The Quadratic Formula
• Completing the Square

In this exercise, we will focus on solving by completing the square.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be written as the square of a binomial. For example, (x + 3)^2 = x^2 + 6x + 9 . When completing the square, we aim to form a perfect square trinomial on one side of the equation. In our exercise, we started with the equation v^2 + 6v = 40. First, we moved the constant term to the other side, isolating the quadratic and linear terms: v^2 + 6v = 40 . Next, we need to determine what number will complete the square. We take the coefficient of the linear term (which is 6), divide by 2 to get 3, and then square it to get 9. We then add and subtract 9 to the left side of the equation, creating a perfect square trinomial: v^2 + 6v + 9 - 9 = 40 . Simplifying gives us v^2 + 6v + 9 = 49 , which can be written as (v + 3)^2 = 49 .

Solving Equations

After forming the perfect square trinomial and rewriting it as (v + 3)^2 = 49 , we can solve for the variable by taking the square root of both sides. This gives us two possible equations: v + 3 = 7 and v + 3 = -7 . Solving these equations separately, we get two solutions:

• For v + 3 = 7: Subtract 3 from both sides to get v = 4 .
• For v + 3 = -7: Subtract 3 from both sides to get v = -10 .

These solutions mean that the original quadratic equation v^2 + 6v = 40 has two solutions: v = 4 and v = -10 . By understanding how to form and solve perfect square trinomials, we can solve quadratic equations more effectively.