Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.