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]
