Solve by completing the square:
x^2 + 7/2x =7
(7/2x is a fraction)
To solve this equation by completing the square, follow these steps:
1. Start with the given equation: x^2 + (7/2)x = 7.
2. Move the constant term (7) to the other side of the equation by subtracting it: x^2 + (7/2)x - 7 = 0.
3. To complete the square, focus on the coefficient of x, which is (7/2). Divide it by 2, then square the result. (7/2) divided by 2 is (7/4), and (7/4) squared is 49/16.
4. Add the result from step 3 to both sides of the equation: x^2 + (7/2)x + 49/16 - 7 = 49/16.
5. Simplify the equation: x^2 + (7/2)x + 49/16 - 112/16 = 0.
6. Combine like terms: x^2 + (7/2)x - 63/16 = 0.
7. Observe that the left side of the equation can be factored as a perfect square trinomial: (x + 7/4)^2 - 63/16 = 0.
8. Move the constant term (63/16) to the other side of the equation: (x + 7/4)^2 = 63/16.
9. Take the square root of both sides of the equation, remembering to consider both the positive and negative square roots: x + 7/4 = ±√(63/16).
10. Simplify the square root on the right side of the equation: x + 7/4 = ±(√63)/4.
11. Move the constant term (7/4) to the other side of the equation: x = -7/4 ± (√63)/4.
So, the solutions to the equation x^2 + (7/2)x = 7 are x = -7/4 + (√63)/4 and x = -7/4 - (√63)/4.