Solve x^2-25x+104=-7x-3

$$x ^ { 2 } - 25 x + 104 = - 7 x - 3$$

Solve for x (complex solution)

$$x=9+\sqrt{26}i\approx 9+5.099019514i$$

$$x=-\sqrt{26}i+9\approx 9-5.099019514i$$

Steps Using the Quadratic Formula

Add 7x to both sides.

$$x^{2}-25x+104+7x=-3$$

Combine -25x and 7x to get -18x.

$$x^{2}-18x+104=-3$$

Add 3 to both sides.

$$x^{2}-18x+104+3=0$$

Add 104 and 3 to get 107.

$$x^{2}-18x+107=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 107 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 107}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 107}}{2}$$

Multiply -4 times 107.

$$x=\frac{-\left(-18\right)±\sqrt{324-428}}{2}$$

Add 324 to -428.

$$x=\frac{-\left(-18\right)±\sqrt{-104}}{2}$$

Take the square root of -104.

$$x=\frac{-\left(-18\right)±2\sqrt{26}i}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{26}i}{2}$$

Now solve the equation x=\frac{18±2\sqrt{26}i}{2} when ± is plus. Add 18 to 2i\sqrt{26}.

$$x=\frac{18+2\sqrt{26}i}{2}$$

Divide 18+2i\sqrt{26} by 2.

$$x=9+\sqrt{26}i$$

Now solve the equation x=\frac{18±2\sqrt{26}i}{2} when ± is minus. Subtract 2i\sqrt{26} from 18.

$$x=\frac{-2\sqrt{26}i+18}{2}$$

Divide 18-2i\sqrt{26} by 2.

$$x=-\sqrt{26}i+9$$

The equation is now solved.

$$x=9+\sqrt{26}i x=-\sqrt{26}i+9$$

Steps for Completing the Square

Add 7x to both sides.

$$x^{2}-25x+104+7x=-3$$

Combine -25x and 7x to get -18x.

$$x^{2}-18x+104=-3$$

Subtract 104 from both sides.

$$x^{2}-18x=-3-104$$

Subtract 104 from -3 to get -107.

$$x^{2}-18x=-107$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-107+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-107+81$$

Add -107 to 81.

$$x^{2}-18x+81=-26$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=-26$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-26}$$

Simplify.

$$x-9=\sqrt{26}i x-9=-\sqrt{26}i$$

Add 9 to both sides of the equation.

$$x=9+\sqrt{26}i x=-\sqrt{26}i+9$$