Solve left(6xright)^2+7x-7=0

$${ \left(6x \right) }^{ 2 } +7x-7 = 0$$

Solve for x

$$x=\frac{\sqrt{1057}-7}{72}\approx 0.354326895$$

$$x=\frac{-\sqrt{1057}-7}{72}\approx -0.548771339$$

Steps Using the Quadratic Formula

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+7x-7=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+7x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 7 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 36\left(-7\right)}}{2\times 36}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 36\left(-7\right)}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-7±\sqrt{49-144\left(-7\right)}}{2\times 36}$$

Multiply -144 times -7.

$$x=\frac{-7±\sqrt{49+1008}}{2\times 36}$$

Add 49 to 1008.

$$x=\frac{-7±\sqrt{1057}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-7±\sqrt{1057}}{72}$$

Now solve the equation x=\frac{-7±\sqrt{1057}}{72} when ± is plus. Add -7 to \sqrt{1057}.

$$x=\frac{\sqrt{1057}-7}{72}$$

Now solve the equation x=\frac{-7±\sqrt{1057}}{72} when ± is minus. Subtract \sqrt{1057} from -7.

$$x=\frac{-\sqrt{1057}-7}{72}$$

The equation is now solved.

$$x=\frac{\sqrt{1057}-7}{72} x=\frac{-\sqrt{1057}-7}{72}$$

Steps for Completing the Square

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+7x-7=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+7x-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$36x^{2}+7x=7$$

Divide both sides by 36.

$$\frac{36x^{2}+7x}{36}=\frac{7}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\frac{7}{36}x=\frac{7}{36}$$

Divide \frac{7}{36}, the coefficient of the x term, by 2 to get \frac{7}{72}. Then add the square of \frac{7}{72} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{36}x+\left(\frac{7}{72}\right)^{2}=\frac{7}{36}+\left(\frac{7}{72}\right)^{2}$$

Square \frac{7}{72} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{36}x+\frac{49}{5184}=\frac{7}{36}+\frac{49}{5184}$$

Add \frac{7}{36} to \frac{49}{5184} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{36}x+\frac{49}{5184}=\frac{1057}{5184}$$

Factor x^{2}+\frac{7}{36}x+\frac{49}{5184}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{72}\right)^{2}=\frac{1057}{5184}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{72}\right)^{2}}=\sqrt{\frac{1057}{5184}}$$

Simplify.

$$x+\frac{7}{72}=\frac{\sqrt{1057}}{72} x+\frac{7}{72}=-\frac{\sqrt{1057}}{72}$$

Subtract \frac{7}{72} from both sides of the equation.

$$x=\frac{\sqrt{1057}-7}{72} x=\frac{-\sqrt{1057}-7}{72}$$