2x^2-x-28=0
Completing the square.
2x^2-x-28 = 0
2(x^2 - 1/2 x) = 28
2(x^2 - 1/2 x + (1/4)^2) = 28 + 2(1/4)^2
2(x - 1/4)^2 = 225/8
(x - 1/4)^2 = 225/16
x - 1/4 = ±15/4
x = 1/4 ± 15/4
x = -7/2, 4
Okay, so I also got the same answer using the Quadratic Formula. Is that okay to happen?
it better happen!
There is only one correct answer...
Of course, the problem said to complete the square.
Well, completing the square is like giving a math problem its own little makeover. It's like saying, "Hey problem, you look a little messy. Let me tidy you up!" So here's what we do:
Step 1: Move the constant term (in this case -28) to the other side of the equation. Now we have 2x^2 - x = 28.
Step 2: Divide both sides of the equation by the coefficient of x^2 (in this case 2). This gives us x^2 - (1/2)x = 14.
Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation. The coefficient of x here is -1/2, so half of it is -1/4, and squaring it gives 1/16. Now our equation becomes x^2 - (1/2)x + 1/16 = 14 + 1/16.
Step 4: Simplify the right side of the equation. 14 + 1/16 is an addition problem you don't want to miss! It equals...wait for it...drum roll please...it's 225/16! Tada!
Now we have transformed our equation into a perfect square! The left side is now (x - 1/4)^2 = 225/16.
And there you have it: the equation, after completing the square, looks all nice, tidy, and ready to solve.
To complete the square for the quadratic equation 2x^2 - x - 28 = 0, follow these steps:
Step 1: Move the constant term to the other side of the equation:
2x^2 - x = 28
Step 2: Divide the coefficient of x^2 by 2 and square the result. Add this value to both sides of the equation:
2(x^2 - (1/2)x) = 28
2(x^2 - (1/2)x + (1/4)*(1/4)) = 28 + 2*(1/4)*(1/4)
2(x^2 - (1/2)x + (1/4)) = 28 + (1/4)
2(x - 1/4)^2 = 113/4
Step 3: Divide both sides of the equation by the coefficient of x^2 to isolate the squared term:
(x - 1/4)^2 = (113/4) / 2
(x - 1/4)^2 = 113/8
Now we have a perfect square trinomial on the left side of the equation.
Step 4: Take the square root of both sides to solve for x:
x - 1/4 = ±√(113/8)
Step 5: Add 1/4 to both sides to isolate x:
x = 1/4 ±√(113/8)
These are the two solutions to the quadratic equation after completing the square.