Use the work shown to find the solutions of the quadratic equation.
x2 – x – StartFraction 3 Over 4 EndFraction = 0
x2 – x = StartFraction 3 Over 4 EndFraction
x2 – x + (StartFraction 1 Over 2 EndFraction) squared = StartFraction 3 Over 4 EndFraction + (StartFraction 1 Over 2 EndFraction) squared
x2 – x + StartFraction 1 Over 4 EndFraction = StartFraction 3 Over 4 EndFraction + StartFraction 1 Over 4 EndFraction
Which is a solution of x2 – x – StartFraction 3 Over 4 EndFraction = 0?
avoid all that gibberish by just typing:
x^2 - x - 3/4 = 0
let's complete the square:
x^2 - x + ... = 3/4 + ...
x^2 - x + 1/4 = 3/4 + 1/4
(x - 1/2)^2 = 1
x - 1/2 = ± 1
x = 1+1/2 or x = -1+1/2
x = 3/2 or x = -1/2
you decide what to do with my answer, (which is correct for the given equation)
To find the solutions of the quadratic equation x^2 - x - 3/4 = 0, we can use the quadratic formula or factorization. Let's use factorization to solve it.
The given equation can be rewritten as:
x^2 - x - 3/4 = 0
To solve this equation, we want to factorize it into two binomial expressions:
(x - a)(x - b) = 0
Where a and b are two values that satisfy the equation. By equating the coefficients of the equation, we can find the values of a and b.
Comparing the equation with (x - a)(x - b) = 0, we get:
a + b = -1
ab = -3/4
Now, we need to find the values of a and b that satisfy these conditions.
By trial and error, we can find that a = -3/4 and b = 1.
So the equation can be factorized as:
(x - 3/4)(x - 1) = 0
Thus, we have two solutions:
x - 3/4 = 0 => x = 3/4
x - 1 = 0 => x = 1
Therefore, the solutions to the quadratic equation x^2 - x - 3/4 = 0 are x = 3/4 and x = 1.
To find the solutions of the quadratic equation x2 – x – 3/4 = 0, we can complete the square.
1. Start with the equation x2 – x = 3/4.
2. Next, take half of the coefficient of the x term, which is -1/2, and square it: (1/2)2 = 1/4.
3. Add this squared term to both sides of the equation: x2 – x + 1/4 = 3/4 + 1/4.
4. Simplify: x2 – x + 1/4 = 4/4.
5. Combine like terms: x2 – x + 1/4 = 1.
6. Rewrite the left side of the equation as a squared binomial: (x - 1/2)2 = 1.
7. Take the square root of both sides: √((x - 1/2)2) = ±√1.
This gives two possible equations:
x - 1/2 = ±1 or x - 1/2 = ±1
8. Solve for x in each equation:
a. x - 1/2 = 1: Add 1/2 to both sides: x = 1 + 1/2 = 3/2
b. x - 1/2 = -1: Add 1/2 to both sides: x = -1 + 1/2 = -1/2
c. x - 1/2 = 1: Subtract 1/2 from both sides: x = 1 - 1/2 = 1/2
d. x - 1/2 = -1: Subtract 1/2 from both sides: x = -1 - 1/2 = -3/2
Therefore, the solutions to the equation x2 – x – 3/4 = 0 are x = 3/2 and x = -1/2.