Problem 42 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}+x+\frac{1}{4}=\frac{9}{16}\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{4}\) and \(x = -\frac{5}{4}\).

Step 1: Move Constant to Right Side

The given equation is \(x^{2} + x + \frac{1}{4} = \frac{9}{16}\). First, we move \(\frac{1}{4}\) to the right side of the equation: \[x^{2} + x = \frac{9}{16} - \frac{1}{4}\].

Step 2: Simplify the Right Side

Subtract \(\frac{1}{4}\) from \(\frac{9}{16}\). Convert \(\frac{1}{4}\) to \(\frac{4}{16}\) to have a common denominator: \[\frac{9}{16} - \frac{4}{16} = \frac{5}{16}\]. Thus, we have \(x^{2} + x = \frac{5}{16}\).

Step 3: Complete the Square

To complete the square on the left side, add \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) to both sides: \[x^{2} + x + \frac{1}{4} = \frac{5}{16} + \frac{1}{4}\].

Step 4: Simplify the Equation

Simplify the right side of the equation by converting \(\frac{1}{4}\) to \(\frac{4}{16}\) and adding: \[\frac{5}{16} + \frac{4}{16} = \frac{9}{16}\]. Now, the equation is \(x^{2} + x + \frac{1}{4} = \frac{9}{16}\).

Step 5: Take the Square Root

The equation is now a perfect square: \((x + \frac{1}{2})^2 = \frac{9}{16}\). Apply the Square Root Property: \[x + \frac{1}{2} = \pm \sqrt{\frac{9}{16}}\].

Step 6: Simplify the Square Root

Calculate \(\sqrt{\frac{9}{16}} = \frac{3}{4}\). Thus, \(x + \frac{1}{2} = \pm \frac{3}{4}\).

Step 7: Solve for x

There are two equations to solve: \(x + \frac{1}{2} = \frac{3}{4}\) and \(x + \frac{1}{2} = -\frac{3}{4}\). Solve each by subtracting \(\frac{1}{2}\): 1) \(x = \frac{3}{4} - \frac{1}{2} = \frac{1}{4}\). 2) \(x = -\frac{3}{4} - \frac{1}{2} = -\frac{5}{4}\).

Step 8: Final Step: State the Solution

The solutions to the equation are \(x = \frac{1}{4}\) and \(x = -\frac{5}{4}\).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. It transforms a quadratic equation into a perfect square trinomial, which is then easier to solve. Here's how it works:
First, you need to ensure the equation is set to zero so you can focus on the left side term that contains the variable. For our exercise, first remove the constant from one side to simplify, bringing the equation to:

• Given: \( x^{2} + x + \frac{1}{4} = \frac{9}{16} \)
• Subtract \(\frac{1}{4}\) from both sides: \( x^{2} + x = \frac{9}{16} - \frac{1}{4} \)

The next step involves making the quadratic expression a perfect square by adding a specific value to both sides. This value is \(\left(\frac{b}{2}\right)^2\), where \( b \) is the coefficient of the linear \( x \) term. Here, \( b=1 \), so you add \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) to form a trinomial:

• Add \(\frac{1}{4}\) to both sides to complete the square: \( x^{2} + x + \frac{1}{4} = \frac{5}{16} + \frac{1}{4} \)
• Solve: \( x^{2} + x + \frac{1}{4} = \frac{9}{16} \)

Solving Quadratic Equations

Solving quadratic equations using the Square Root Property involves creating a perfect square trinomial. Once the equation resembles the form of \( (x+c)^2 = d \), you can easily solve for the variable by taking the square root of both sides.
For this method, first transform the equation to the perfect square form. In our exercise, this is achieved when

• Equation is: \( (x + \frac{1}{2})^2 = \frac{9}{16} \)

Then, apply the Square Root Property, providing a solution in two parts: one for the positive square root and one for the negative square root. This is because a squared value can come from both a positive and negative base. This requires setting \( x + \frac{1}{2} \) equal to \(\frac{3}{4}\) and \(-\frac{3}{4}\):

• \( x + \frac{1}{2} = \pm \frac{3}{4} \)
• Subtract \( \frac{1}{2} \) from both solutions:
  • \( x = \frac{1}{4} \)
  • \( x = -\frac{5}{4} \)

These steps reveal the values of \( x \) that satisfy the original equation.

Perfect Square Trinomial

A perfect square trinomial forms when a quadratic expression can be rewritten as the square of a binomial. This crucial concept simplifies both completing the square and solving with the Square Root Property.
The idea is to take a quadratic expression \( ax^2 + bx + c \) and reformat it to \( (x + d)^2 \). For this conversion:

• Ensure the coefficient of \( x^2 \) is 1. In our case, it already is.
• Half the linear term's coefficient: \( b = 1 \) means \( \frac{1}{2} \).
• Square the result: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \), so the perfect square trinomial is \( (x + \frac{1}{2})^2 \).

This trinomial representation means any equation structured in this form can be tackled using the Square Root Property. Once formatted correctly, solving becomes straightforward. Our exercise shows this process and illustrates how algebraic manipulation turns an equation to an easily solvable format, further driving home the utility of mastering this skill.