Question

1. $-x^{2}+2x=7$

Explanation:

Step1: Rewrite to standard quadratic form

Multiply both sides by -1, then rearrange terms:
$$x^2 - 2x + 7 = 0$$

Step2: Identify coefficients for quadratic formula

For $ax^2+bx+c=0$, $a=1$, $b=-2$, $c=7$

Step3: Calculate discriminant

$$\Delta = b^2 - 4ac = (-2)^2 - 4(1)(7) = 4 - 28 = -24$$

Step4: Apply quadratic formula

Since $\Delta < 0$, use complex roots: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$
$$x = \frac{2 \pm \sqrt{-24}}{2} = \frac{2 \pm 2i\sqrt{6}}{2}$$

Step5: Simplify the expression

$$x = 1 \pm i\sqrt{6}$$

Answer:

$x = 1 + i\sqrt{6}$ and $x = 1 - i\sqrt{6}$