Solve x^2+67=18x

$$x ^ { 2 } + 67 = 18 x$$

Solve for x

$$x=\sqrt{14}+9\approx 12.741657387$$

$$x=9-\sqrt{14}\approx 5.258342613$$

Steps Using the Quadratic Formula

Subtract 18x from both sides.

$$x^{2}+67-18x=0$$

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.

$$x^{2}-18x+67=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 67 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 67}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 67}}{2}$$

Multiply -4 times 67.

$$x=\frac{-\left(-18\right)±\sqrt{324-268}}{2}$$

Add 324 to -268.

$$x=\frac{-\left(-18\right)±\sqrt{56}}{2}$$

Take the square root of 56.

$$x=\frac{-\left(-18\right)±2\sqrt{14}}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{14}}{2}$$

Now solve the equation x=\frac{18±2\sqrt{14}}{2} when ± is plus. Add 18 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}+18}{2}$$

Divide 18+2\sqrt{14} by 2.

$$x=\sqrt{14}+9$$

Now solve the equation x=\frac{18±2\sqrt{14}}{2} when ± is minus. Subtract 2\sqrt{14} from 18.

$$x=\frac{18-2\sqrt{14}}{2}$$

Divide 18-2\sqrt{14} by 2.

$$x=9-\sqrt{14}$$

The equation is now solved.

$$x=\sqrt{14}+9 x=9-\sqrt{14}$$

Steps for Completing the Square

Subtract 18x from both sides.

$$x^{2}+67-18x=0$$

Subtract 67 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-18x=-67$$

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.

$$x^{2}-18x+\left(-9\right)^{2}=-67+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-67+81$$

Add -67 to 81.

$$x^{2}-18x+81=14$$

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}.

$$\left(x-9\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{14}$$

Simplify.

$$x-9=\sqrt{14} x-9=-\sqrt{14}$$

Add 9 to both sides of the equation.

$$x=\sqrt{14}+9 x=9-\sqrt{14}$$