Solve 100=1-2x+x^2+14

$$100 = 1 - 2 x + x ^ { 2 } + 14$$

Solve for x

$$x=\sqrt{86}+1\approx 10.273618495$$

$$x=1-\sqrt{86}\approx -8.273618495$$

Steps Using the Quadratic Formula

Add 1 and 14 to get 15.

$$100=15-2x+x^{2}$$

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

$$15-2x+x^{2}=100$$

Subtract 100 from both sides.

$$15-2x+x^{2}-100=0$$

Subtract 100 from 15 to get -85.

$$-85-2x+x^{2}=0$$

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.

$$x^{2}-2x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-85\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-85\right)}}{2}$$

Multiply -4 times -85.

$$x=\frac{-\left(-2\right)±\sqrt{4+340}}{2}$$

Add 4 to 340.

$$x=\frac{-\left(-2\right)±\sqrt{344}}{2}$$

Take the square root of 344.

$$x=\frac{-\left(-2\right)±2\sqrt{86}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{86}}{2}$$

Now solve the equation x=\frac{2±2\sqrt{86}}{2} when ± is plus. Add 2 to 2\sqrt{86}.

$$x=\frac{2\sqrt{86}+2}{2}$$

Divide 2+2\sqrt{86} by 2.

$$x=\sqrt{86}+1$$

Now solve the equation x=\frac{2±2\sqrt{86}}{2} when ± is minus. Subtract 2\sqrt{86} from 2.

$$x=\frac{2-2\sqrt{86}}{2}$$

Divide 2-2\sqrt{86} by 2.

$$x=1-\sqrt{86}$$

The equation is now solved.

$$x=\sqrt{86}+1 x=1-\sqrt{86}$$

Steps for Completing the Square

Add 1 and 14 to get 15.

$$100=15-2x+x^{2}$$

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

$$15-2x+x^{2}=100$$

Subtract 15 from both sides.

$$-2x+x^{2}=100-15$$

Subtract 15 from 100 to get 85.

$$-2x+x^{2}=85$$

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.

$$x^{2}-2x=85$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=85+1$$

Add 85 to 1.

$$x^{2}-2x+1=86$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=86$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{86}$$

Simplify.

$$x-1=\sqrt{86} x-1=-\sqrt{86}$$

Add 1 to both sides of the equation.

$$x=\sqrt{86}+1 x=1-\sqrt{86}$$