Solve x^2-16x+50=21

$$x ^ { 2 } - 16 x + 50 = 21$$

Solve for x

$$x=\sqrt{35}+8\approx 13.916079783$$

$$x=8-\sqrt{35}\approx 2.083920217$$

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.

$$x^{2}-16x+50=21$$

Subtract 21 from both sides of the equation.

$$x^{2}-16x+50-21=21-21$$

Subtracting 21 from itself leaves 0.

$$x^{2}-16x+50-21=0$$

Subtract 21 from 50.

$$x^{2}-16x+29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 29}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 29}}{2}$$

Multiply -4 times 29.

$$x=\frac{-\left(-16\right)±\sqrt{256-116}}{2}$$

Add 256 to -116.

$$x=\frac{-\left(-16\right)±\sqrt{140}}{2}$$

Take the square root of 140.

$$x=\frac{-\left(-16\right)±2\sqrt{35}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{35}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{35}}{2} when ± is plus. Add 16 to 2\sqrt{35}.

$$x=\frac{2\sqrt{35}+16}{2}$$

Divide 16+2\sqrt{35} by 2.

$$x=\sqrt{35}+8$$

Now solve the equation x=\frac{16±2\sqrt{35}}{2} when ± is minus. Subtract 2\sqrt{35} from 16.

$$x=\frac{16-2\sqrt{35}}{2}$$

Divide 16-2\sqrt{35} by 2.

$$x=8-\sqrt{35}$$

The equation is now solved.

$$x=\sqrt{35}+8 x=8-\sqrt{35}$$

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.

$$x^{2}-16x+50=21$$

Subtract 50 from both sides of the equation.

$$x^{2}-16x+50-50=21-50$$

Subtracting 50 from itself leaves 0.

$$x^{2}-16x=21-50$$

Subtract 50 from 21.

$$x^{2}-16x=-29$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-29+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-29+64$$

Add -29 to 64.

$$x^{2}-16x+64=35$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=35$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{35}$$

Simplify.

$$x-8=\sqrt{35} x-8=-\sqrt{35}$$

Add 8 to both sides of the equation.

$$x=\sqrt{35}+8 x=8-\sqrt{35}$$