Solve x^2-22x+65=0

$${ x }^{ 2 } -22x+65=0$$

Solve for x

$$x=2\sqrt{14}+11\approx 18.483314774$$

$$x=11-2\sqrt{14}\approx 3.516685226$$

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.

$$x^{2}-22x+65=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -22 for b, and 65 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 65}}{2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 65}}{2}$$

Multiply -4 times 65.

$$x=\frac{-\left(-22\right)±\sqrt{484-260}}{2}$$

Add 484 to -260.

$$x=\frac{-\left(-22\right)±\sqrt{224}}{2}$$

Take the square root of 224.

$$x=\frac{-\left(-22\right)±4\sqrt{14}}{2}$$

The opposite of -22 is 22.

$$x=\frac{22±4\sqrt{14}}{2}$$

Now solve the equation x=\frac{22±4\sqrt{14}}{2} when ± is plus. Add 22 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}+22}{2}$$

Divide 22+4\sqrt{14} by 2.

$$x=2\sqrt{14}+11$$

Now solve the equation x=\frac{22±4\sqrt{14}}{2} when ± is minus. Subtract 4\sqrt{14} from 22.

$$x=\frac{22-4\sqrt{14}}{2}$$

Divide 22-4\sqrt{14} by 2.

$$x=11-2\sqrt{14}$$

The equation is now solved.

$$x=2\sqrt{14}+11 x=11-2\sqrt{14}$$

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.

$$x^{2}-22x+65=0$$

Subtract 65 from both sides of the equation.

$$x^{2}-22x+65-65=-65$$

Subtracting 65 from itself leaves 0.

$$x^{2}-22x=-65$$

Divide -22, the coefficient of the x term, by 2 to get -11. Then add the square of -11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-22x+\left(-11\right)^{2}=-65+\left(-11\right)^{2}$$

Square -11.

$$x^{2}-22x+121=-65+121$$

Add -65 to 121.

$$x^{2}-22x+121=56$$

Factor x^{2}-22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-11\right)^{2}=56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{56}$$

Simplify.

$$x-11=2\sqrt{14} x-11=-2\sqrt{14}$$

Add 11 to both sides of the equation.

$$x=2\sqrt{14}+11 x=11-2\sqrt{14}$$