Solve 1=x^2-8x+15

$$1 = x ^ { 2 } - 8 x + 15$$

Solve for x

$$x=\sqrt{2}+4\approx 5.414213562$$

$$x=4-\sqrt{2}\approx 2.585786438$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-8x+15=1$$

Subtract 1 from both sides.

$$x^{2}-8x+15-1=0$$

Subtract 1 from 15 to get 14.

$$x^{2}-8x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 14}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 14}}{2}$$

Multiply -4 times 14.

$$x=\frac{-\left(-8\right)±\sqrt{64-56}}{2}$$

Add 64 to -56.

$$x=\frac{-\left(-8\right)±\sqrt{8}}{2}$$

Take the square root of 8.

$$x=\frac{-\left(-8\right)±2\sqrt{2}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{2}}{2}$$

Now solve the equation x=\frac{8±2\sqrt{2}}{2} when ± is plus. Add 8 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+8}{2}$$

Divide 2\sqrt{2}+8 by 2.

$$x=\sqrt{2}+4$$

Now solve the equation x=\frac{8±2\sqrt{2}}{2} when ± is minus. Subtract 2\sqrt{2} from 8.

$$x=\frac{8-2\sqrt{2}}{2}$$

Divide 8-2\sqrt{2} by 2.

$$x=4-\sqrt{2}$$

The equation is now solved.

$$x=\sqrt{2}+4 x=4-\sqrt{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-8x+15=1$$

Subtract 15 from both sides.

$$x^{2}-8x=1-15$$

Subtract 15 from 1 to get -14.

$$x^{2}-8x=-14$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-14+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-14+16$$

Add -14 to 16.

$$x^{2}-8x+16=2$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=2$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{2}$$

Simplify.

$$x-4=\sqrt{2} x-4=-\sqrt{2}$$

Add 4 to both sides of the equation.

$$x=\sqrt{2}+4 x=4-\sqrt{2}$$