How do you solve #y − x^2 = −6x − 7#?

Question

Eli Assouline · Accepted Answer

Solve : y = x^2 - 6x - 7 = 0 Apply the shortcut rule.
Since (a - b + c) = 0, one real root is (-1) and the other is (-c/a = 7)

Savannah Anderson · Answer

undefined To solve the equation $y - x^2 = -6x - 7$:

1. Rewrite the equation in the form $ax^2 + bx + c = 0$, which becomes $x^2 + 6x + y + 7 = 0$ since we want $x$ terms on one side and constant terms on the other.

2. Rearrange the terms to get $x^2 + 6x + (y + 7) = 0$.

3. Apply the quadratic formula: $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$.

4. Substitute the values: $x = \frac{{-6 \pm \sqrt{{6^2 - 4 \cdot 1 \cdot (y + 7)}}}}{{2 \cdot 1}}$.

5. Simplify: $x = \frac{{-6 \pm \sqrt{{36 - 4(y + 7)}}}}{2}$.

6. Simplify further: $x = \frac{{-6 \pm \sqrt{{36 - 4y - 28}}}}{2}$.

7. Simplify inside the square root: $x = \frac{{-6 \pm \sqrt{{8 - 4y}}}}{2}$.

8. Further simplify if possible: $x = \frac{{-6 \pm \sqrt{{4(2 - y)}}}}{2}$.

9. Factor out the 4 from the square root: $x = \frac{{-6 \pm 2\sqrt{{2 - y}}}}{2}$.

10. Simplify: $x = -3 \pm \sqrt{{2 - y}}$.

Therefore, the solutions to the equation $y - x^2 = -6x - 7$ are $x = -3 + \sqrt{{2 - y}}$ and $x = -3 - \sqrt{{2 - y}}$.