3. Solve for x in 8x^2 + 2x - 4 = 0
I have tried factoring, taking out a GCF then factoring, completing the square, and got a bunch of wrong answers. I thought we were supposed to complete the square, so maybe I'm just doing that wrong. I'm not looking for the answer to the problem - just a way to solve it.
Here's what I tried to do for completing the square:
i. (b/2)^2 = (2/2)^2 = 1
ii. 8x^2 + 2x + 1 = 5
iii. what? It's not a perfect square.
You'll need to use the quadratic formula on that one. Or, completing the square,
8x^2+2x-4 = 0
8(x^2 + x/4) = 4
8(x^2 + x/4 + 1/64) = 4 + 8/64
8(x + 1/8)^2 = 4 + 1/8
(x + 1/8)^2 = 33/64
x + 1/8 = ±√33 / 8
x = -1/8 ±√33/8
x = (-1±√33)/8
To solve the quadratic equation 8x^2 + 2x - 4 = 0, completing the square is a valid method. Let me guide you step-by-step through the process:
1. Start with the quadratic equation: 8x^2 + 2x - 4 = 0.
2. First, divide the entire equation by the coefficient of x^2, which is 8. This step will simplify the equation: x^2 + (2/8)x - (4/8) = 0.
3. Simplifying further, we get: x^2 + (1/4)x - (1/2) = 0.
4. Now, we want to complete the square. To do this, we take half the coefficient of x and square it. In this equation, the coefficient of x is (1/4). Half of (1/4) is (1/8), and when squared, it becomes (1/64).
5. Add (1/64) to both sides of the equation. This maintains the equality: x^2 + (1/4)x - (1/2) + (1/64) = (1/64).
6. To simplify the left side of the equation, factor the quadratic expression. In this case, it factors to: (x + 1/8)^2 - (33/64) = (1/64).
7. Next, move the constant term to the right side of the equation: (x + 1/8)^2 = (1/64) + (33/64).
8. Combining the fractions on the right side, we have: (x + 1/8)^2 = (34/64).
9. Simplify the fractions: (x + 1/8)^2 = (17/32).
10. Now, take the square root of both sides of the equation: √[(x + 1/8)^2] = ±√(17/32).
11. Removing the square root on the left side gives: x + 1/8 = ±√(17/32).
12. Subtract 1/8 from both sides: x = -1/8 ±√(17/32).
13. Finally, simplify the expression by multiplying the fraction: x = (-1 ± 2√17) / 8.
Therefore, the solution to the equation 8x^2 + 2x - 4 = 0 is x = (-1 ± 2√17) / 8.