Solve 2x^2-34x=-22

$$2 x ^ { 2 } - 34 x = - 22$$

Solve for x

$$x = \frac{7 \sqrt{5} + 17}{2} \approx 16.326237921$$

$$x=\frac{17-7\sqrt{5}}{2}\approx 0.673762079$$

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.

$$2x^{2}-34x=-22$$

Add 22 to both sides of the equation.

$$2x^{2}-34x-\left(-22\right)=-22-\left(-22\right)$$

Subtracting -22 from itself leaves 0.

$$2x^{2}-34x-\left(-22\right)=0$$

Subtract -22 from 0.

$$2x^{2}-34x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -34 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 22}}{2\times 2}$$

Square -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 22}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 22}}{2\times 2}$$

Multiply -8 times 22.

$$x=\frac{-\left(-34\right)±\sqrt{1156-176}}{2\times 2}$$

Add 1156 to -176.

$$x=\frac{-\left(-34\right)±\sqrt{980}}{2\times 2}$$

Take the square root of 980.

$$x=\frac{-\left(-34\right)±14\sqrt{5}}{2\times 2}$$

The opposite of -34 is 34.

$$x=\frac{34±14\sqrt{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{34±14\sqrt{5}}{4}$$

Now solve the equation x=\frac{34±14\sqrt{5}}{4} when ± is plus. Add 34 to 14\sqrt{5}.

$$x=\frac{14\sqrt{5}+34}{4}$$

Divide 34+14\sqrt{5} by 4.

$$x=\frac{7\sqrt{5}+17}{2}$$

Now solve the equation x=\frac{34±14\sqrt{5}}{4} when ± is minus. Subtract 14\sqrt{5} from 34.

$$x=\frac{34-14\sqrt{5}}{4}$$

Divide 34-14\sqrt{5} by 4.

$$x=\frac{17-7\sqrt{5}}{2}$$

The equation is now solved.

$$x=\frac{7\sqrt{5}+17}{2} x=\frac{17-7\sqrt{5}}{2}$$

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.

$$2x^{2}-34x=-22$$

Divide both sides by 2.

$$\frac{2x^{2}-34x}{2}=-\frac{22}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{22}{2}$$

Divide -34 by 2.

$$x^{2}-17x=-\frac{22}{2}$$

Divide -22 by 2.

$$x^{2}-17x=-11$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-11+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-11+\frac{289}{4}$$

Add -11 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{245}{4}$$

Factor x^{2}-17x+\frac{289}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{245}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{245}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{7\sqrt{5}}{2} x-\frac{17}{2}=-\frac{7\sqrt{5}}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=\frac{7\sqrt{5}+17}{2} x=\frac{17-7\sqrt{5}}{2}$$