Solve left(6xright)^2++130x+9=0

$${ \left(6x \right) }^{ 2 } ++130x+9 = 0$$

Solve for x

$$x=\frac{\sqrt{3901}-65}{36}\approx -0.070611503$$

$$x=\frac{-\sqrt{3901}-65}{36}\approx -3.540499608$$

Steps Using the Quadratic Formula

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

$$6^{2}x^{2}+130x+9=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+130x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 130 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-130±\sqrt{130^{2}-4\times 36\times 9}}{2\times 36}$$

Square 130.

$$x=\frac{-130±\sqrt{16900-4\times 36\times 9}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-130±\sqrt{16900-144\times 9}}{2\times 36}$$

Multiply -144 times 9.

$$x=\frac{-130±\sqrt{16900-1296}}{2\times 36}$$

Add 16900 to -1296.

$$x=\frac{-130±\sqrt{15604}}{2\times 36}$$

Take the square root of 15604.

$$x=\frac{-130±2\sqrt{3901}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-130±2\sqrt{3901}}{72}$$

Now solve the equation x=\frac{-130±2\sqrt{3901}}{72} when ± is plus. Add -130 to 2\sqrt{3901}.

$$x=\frac{2\sqrt{3901}-130}{72}$$

Divide -130+2\sqrt{3901} by 72.

$$x=\frac{\sqrt{3901}-65}{36}$$

Now solve the equation x=\frac{-130±2\sqrt{3901}}{72} when ± is minus. Subtract 2\sqrt{3901} from -130.

$$x=\frac{-2\sqrt{3901}-130}{72}$$

Divide -130-2\sqrt{3901} by 72.

$$x=\frac{-\sqrt{3901}-65}{36}$$

The equation is now solved.

$$x=\frac{\sqrt{3901}-65}{36} x=\frac{-\sqrt{3901}-65}{36}$$

Steps for Completing the Square

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

$$6^{2}x^{2}+130x+9=0$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+130x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$36x^{2}+130x=-9$$

Divide both sides by 36.

$$\frac{36x^{2}+130x}{36}=-\frac{9}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\frac{130}{36}x=-\frac{9}{36}$$

Reduce the fraction \frac{130}{36} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{65}{18}x=-\frac{9}{36}$$

Reduce the fraction \frac{-9}{36} to lowest terms by extracting and canceling out 9.

$$x^{2}+\frac{65}{18}x=-\frac{1}{4}$$

Divide \frac{65}{18}, the coefficient of the x term, by 2 to get \frac{65}{36}. Then add the square of \frac{65}{36} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{65}{18}x+\left(\frac{65}{36}\right)^{2}=-\frac{1}{4}+\left(\frac{65}{36}\right)^{2}$$

Square \frac{65}{36} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{65}{18}x+\frac{4225}{1296}=-\frac{1}{4}+\frac{4225}{1296}$$

Add -\frac{1}{4} to \frac{4225}{1296} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{65}{18}x+\frac{4225}{1296}=\frac{3901}{1296}$$

Factor x^{2}+\frac{65}{18}x+\frac{4225}{1296}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{65}{36}\right)^{2}=\frac{3901}{1296}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{65}{36}\right)^{2}}=\sqrt{\frac{3901}{1296}}$$

Simplify.

$$x+\frac{65}{36}=\frac{\sqrt{3901}}{36} x+\frac{65}{36}=-\frac{\sqrt{3901}}{36}$$

Subtract \frac{65}{36} from both sides of the equation.

$$x=\frac{\sqrt{3901}-65}{36} x=\frac{-\sqrt{3901}-65}{36}$$