Solve x^2+7x=x-10

$$x ^ { 2 } + 7 x = x - 10$$

Solve for x (complex solution)

$$x=-3+i$$

$$x=-3-i$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}+7x-x=-10$$

Combine 7x and -x to get 6x.

$$x^{2}+6x=-10$$

Add 10 to both sides.

$$x^{2}+6x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 10}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-6±\sqrt{36-40}}{2}$$

Add 36 to -40.

$$x=\frac{-6±\sqrt{-4}}{2}$$

Take the square root of -4.

$$x=\frac{-6±2i}{2}$$

Now solve the equation x=\frac{-6±2i}{2} when ± is plus. Add -6 to 2i.

$$x=\frac{-6+2i}{2}$$

Divide -6+2i by 2.

$$x=-3+i$$

Now solve the equation x=\frac{-6±2i}{2} when ± is minus. Subtract 2i from -6.

$$x=\frac{-6-2i}{2}$$

Divide -6-2i by 2.

$$x=-3-i$$

The equation is now solved.

$$x=-3+i x=-3-i$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}+7x-x=-10$$

Combine 7x and -x to get 6x.

$$x^{2}+6x=-10$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=-10+3^{2}$$

Square 3.

$$x^{2}+6x+9=-10+9$$

Add -10 to 9.

$$x^{2}+6x+9=-1$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x+3=i x+3=-i$$

Subtract 3 from both sides of the equation.

$$x=-3+i x=-3-i$$