Solve left(7xright)^2+12x-6=0

$${ \left(7x \right) }^{ 2 } +12x-6 = 0$$

Solve for x

$$x=\frac{\sqrt{330}-6}{49}\approx 0.248283717$$

$$x=\frac{-\sqrt{330}-6}{49}\approx -0.493181676$$

Steps Using the Quadratic Formula

Expand \left(7x\right)^{2}.

$$7^{2}x^{2}+12x-6=0$$

Calculate 7 to the power of 2 and get 49.

$$49x^{2}+12x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, 12 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 49\left(-6\right)}}{2\times 49}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 49\left(-6\right)}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-12±\sqrt{144-196\left(-6\right)}}{2\times 49}$$

Multiply -196 times -6.

$$x=\frac{-12±\sqrt{144+1176}}{2\times 49}$$

Add 144 to 1176.

$$x=\frac{-12±\sqrt{1320}}{2\times 49}$$

Take the square root of 1320.

$$x=\frac{-12±2\sqrt{330}}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{-12±2\sqrt{330}}{98}$$

Now solve the equation x=\frac{-12±2\sqrt{330}}{98} when ± is plus. Add -12 to 2\sqrt{330}.

$$x=\frac{2\sqrt{330}-12}{98}$$

Divide -12+2\sqrt{330} by 98.

$$x=\frac{\sqrt{330}-6}{49}$$

Now solve the equation x=\frac{-12±2\sqrt{330}}{98} when ± is minus. Subtract 2\sqrt{330} from -12.

$$x=\frac{-2\sqrt{330}-12}{98}$$

Divide -12-2\sqrt{330} by 98.

$$x=\frac{-\sqrt{330}-6}{49}$$

The equation is now solved.

$$x=\frac{\sqrt{330}-6}{49} x=\frac{-\sqrt{330}-6}{49}$$

Steps for Completing the Square

Expand \left(7x\right)^{2}.

$$7^{2}x^{2}+12x-6=0$$

Calculate 7 to the power of 2 and get 49.

$$49x^{2}+12x-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$49x^{2}+12x=6$$

Divide both sides by 49.

$$\frac{49x^{2}+12x}{49}=\frac{6}{49}$$

Dividing by 49 undoes the multiplication by 49.

$$x^{2}+\frac{12}{49}x=\frac{6}{49}$$

Divide \frac{12}{49}, the coefficient of the x term, by 2 to get \frac{6}{49}. Then add the square of \frac{6}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{49}x+\left(\frac{6}{49}\right)^{2}=\frac{6}{49}+\left(\frac{6}{49}\right)^{2}$$

Square \frac{6}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{49}x+\frac{36}{2401}=\frac{6}{49}+\frac{36}{2401}$$

Add \frac{6}{49} to \frac{36}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{12}{49}x+\frac{36}{2401}=\frac{330}{2401}$$

Factor x^{2}+\frac{12}{49}x+\frac{36}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{49}\right)^{2}=\frac{330}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{49}\right)^{2}}=\sqrt{\frac{330}{2401}}$$

Simplify.

$$x+\frac{6}{49}=\frac{\sqrt{330}}{49} x+\frac{6}{49}=-\frac{\sqrt{330}}{49}$$

Subtract \frac{6}{49} from both sides of the equation.

$$x=\frac{\sqrt{330}-6}{49} x=\frac{-\sqrt{330}-6}{49}$$