Solve 22x^2-2x=7

$$22 x ^ { 2 } - 2 x = 7$$

Solve for x

$$x=\frac{\sqrt{155}+1}{22}\approx 0.611359073$$

$$x=\frac{1-\sqrt{155}}{22}\approx -0.520449982$$

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.

$$22x^{2}-2x=7$$

Subtract 7 from both sides of the equation.

$$22x^{2}-2x-7=7-7$$

Subtracting 7 from itself leaves 0.

$$22x^{2}-2x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 22 for a, -2 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 22\left(-7\right)}}{2\times 22}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 22\left(-7\right)}}{2\times 22}$$

Multiply -4 times 22.

$$x=\frac{-\left(-2\right)±\sqrt{4-88\left(-7\right)}}{2\times 22}$$

Multiply -88 times -7.

$$x=\frac{-\left(-2\right)±\sqrt{4+616}}{2\times 22}$$

Add 4 to 616.

$$x=\frac{-\left(-2\right)±\sqrt{620}}{2\times 22}$$

Take the square root of 620.

$$x=\frac{-\left(-2\right)±2\sqrt{155}}{2\times 22}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{155}}{2\times 22}$$

Multiply 2 times 22.

$$x=\frac{2±2\sqrt{155}}{44}$$

Now solve the equation x=\frac{2±2\sqrt{155}}{44} when ± is plus. Add 2 to 2\sqrt{155}.

$$x=\frac{2\sqrt{155}+2}{44}$$

Divide 2+2\sqrt{155} by 44.

$$x=\frac{\sqrt{155}+1}{22}$$

Now solve the equation x=\frac{2±2\sqrt{155}}{44} when ± is minus. Subtract 2\sqrt{155} from 2.

$$x=\frac{2-2\sqrt{155}}{44}$$

Divide 2-2\sqrt{155} by 44.

$$x=\frac{1-\sqrt{155}}{22}$$

The equation is now solved.

$$x=\frac{\sqrt{155}+1}{22} x=\frac{1-\sqrt{155}}{22}$$

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.

$$22x^{2}-2x=7$$

Divide both sides by 22.

$$\frac{22x^{2}-2x}{22}=\frac{7}{22}$$

Dividing by 22 undoes the multiplication by 22.

$$x^{2}+\left(-\frac{2}{22}\right)x=\frac{7}{22}$$

Reduce the fraction \frac{-2}{22} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{11}x=\frac{7}{22}$$

Divide -\frac{1}{11}, the coefficient of the x term, by 2 to get -\frac{1}{22}. Then add the square of -\frac{1}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{11}x+\left(-\frac{1}{22}\right)^{2}=\frac{7}{22}+\left(-\frac{1}{22}\right)^{2}$$

Square -\frac{1}{22} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{11}x+\frac{1}{484}=\frac{7}{22}+\frac{1}{484}$$

Add \frac{7}{22} to \frac{1}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{11}x+\frac{1}{484}=\frac{155}{484}$$

Factor x^{2}-\frac{1}{11}x+\frac{1}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{22}\right)^{2}=\frac{155}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{22}\right)^{2}}=\sqrt{\frac{155}{484}}$$

Simplify.

$$x-\frac{1}{22}=\frac{\sqrt{155}}{22} x-\frac{1}{22}=-\frac{\sqrt{155}}{22}$$

Add \frac{1}{22} to both sides of the equation.

$$x=\frac{\sqrt{155}+1}{22} x=\frac{1-\sqrt{155}}{22}$$