Solve x^2-14x+118=5

$$x ^ { 2 } - 14 x + 118 = 5$$

Solve for x (complex solution)

$$x=7+8i$$

$$x=7-8i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-14x+118=5$$

Subtract 5 from both sides of the equation.

$$x^{2}-14x+118-5=5-5$$

Subtracting 5 from itself leaves 0.

$$x^{2}-14x+118-5=0$$

Subtract 5 from 118.

$$x^{2}-14x+113=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 113 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 113}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 113}}{2}$$

Multiply -4 times 113.

$$x=\frac{-\left(-14\right)±\sqrt{196-452}}{2}$$

Add 196 to -452.

$$x=\frac{-\left(-14\right)±\sqrt{-256}}{2}$$

Take the square root of -256.

$$x=\frac{-\left(-14\right)±16i}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±16i}{2}$$

Now solve the equation x=\frac{14±16i}{2} when ± is plus. Add 14 to 16i.

$$x=\frac{14+16i}{2}$$

Divide 14+16i by 2.

$$x=7+8i$$

Now solve the equation x=\frac{14±16i}{2} when ± is minus. Subtract 16i from 14.

$$x=\frac{14-16i}{2}$$

Divide 14-16i by 2.

$$x=7-8i$$

The equation is now solved.

$$x=7+8i x=7-8i$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-14x+118=5$$

Subtract 118 from both sides of the equation.

$$x^{2}-14x+118-118=5-118$$

Subtracting 118 from itself leaves 0.

$$x^{2}-14x=5-118$$

Subtract 118 from 5.

$$x^{2}-14x=-113$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-113+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-113+49$$

Add -113 to 49.

$$x^{2}-14x+49=-64$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=-64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-64}$$

Simplify.

$$x-7=8i x-7=-8i$$

Add 7 to both sides of the equation.

$$x=7+8i x=7-8i$$