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Use the work shown to find the solutions of the quadratic equation.

[tex]\[
\begin{array}{l}
x^2 - x - \frac{3}{4} = 0 \\
x^2 - x = \frac{3}{4} \\
x^2 - x + \left( \frac{1}{2} \right)^2 = \frac{3}{4} + \left( \frac{1}{2} \right)^2 \\
x^2 - x + \frac{1}{4} = \frac{3}{4} + \frac{1}{4} \\
\left( x - \frac{1}{2} \right)^2 = 1
\end{array}
\][/tex]

Which is a solution of [tex]\( x^2 - x - \frac{3}{4} = 0 \)[/tex]?

A. [tex]\(-\frac{1}{4}\)[/tex]
B. [tex]\(\frac{1}{2}\)[/tex]
C. [tex]\(\frac{3}{2}\)[/tex]
D. [tex]\(\frac{3}{4}\)[/tex]

Sagot :

GinnyAnswer
To find the solutions of the quadratic equation [tex]\( x^2 - x - \frac{3}{4} = 0 \)[/tex]:

1. Rewrite the equation:
[tex]\[ x^2 - x - \frac{3}{4} = 0 \][/tex]

2. Complete the square:
Start by moving the constant term to the right side:
[tex]\[ x^2 - x = \frac{3}{4} \][/tex]

Add [tex]\(\left(\frac{1}{2}\right)^2\)[/tex] to both sides to complete the square. Note that the term [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex]:
[tex]\[ x^2 - x + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \left(\frac{1}{2}\right)^2 \][/tex]

Simplify the right-hand side:
[tex]\[ x^2 - x + \frac{1}{4} = \frac{3}{4} + \frac{1}{4} \][/tex]
[tex]\[ x^2 - x + \frac{1}{4} = 1 \][/tex]

3. Factor the left side:
[tex]\[ \left( x - \frac{1}{2} \right)^2 = 1 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[ x - \frac{1}{2} = \pm 1 \][/tex]

This gives us two solutions:
[tex]\[ x - \frac{1}{2} = 1 \quad \text{or} \quad x - \frac{1}{2} = -1 \][/tex]

Simplify each equation:
[tex]\[ x = 1 + \frac{1}{2} \quad \text{or} \quad x = -1 + \frac{1}{2} \][/tex]
[tex]\[ x = \frac{3}{2} \quad \text{or} \quad x = -\frac{1}{2} \][/tex]

So, the solutions to the equation [tex]\( x^2 - x - \frac{3}{4} = 0 \)[/tex] are:
[tex]\[ \boxed{\frac{3}{2}} \quad \text{and} \quad \boxed{-\frac{1}{2}} \][/tex]

Among the given choices, [tex]\(\boxed{\frac{3}{2}}\)[/tex] is indeed one of the solutions.

The correct answer is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
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