Solve x^2+8x^2=13^2

$${ x }^{ 2 } +8 { x }^{ 2 } = { 13 }^{ 2 }$$

Solve for x

$$x = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

$$x = -\frac{13}{3} = -4\frac{1}{3} \approx -4.333333333$$

Steps Using Difference of Squares

Combine x^{2} and 8x^{2} to get 9x^{2}.

$$9x^{2}=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$9x^{2}=169$$

Subtract 169 from both sides.

$$9x^{2}-169=0$$

Consider 9x^{2}-169. Rewrite 9x^{2}-169 as \left(3x\right)^{2}-13^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(3x-13\right)\left(3x+13\right)=0$$

To find equation solutions, solve 3x-13=0 and 3x+13=0.

$$x=\frac{13}{3} x=-\frac{13}{3}$$

Steps by Finding Square Root

Combine x^{2} and 8x^{2} to get 9x^{2}.

$$9x^{2}=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$9x^{2}=169$$

Divide both sides by 9.

$$x^{2}=\frac{169}{9}$$

Take the square root of both sides of the equation.

$$x=\frac{13}{3} x=-\frac{13}{3}$$

Steps Using the Quadratic Formula

Combine x^{2} and 8x^{2} to get 9x^{2}.

$$9x^{2}=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$9x^{2}=169$$

Subtract 169 from both sides.

$$9x^{2}-169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 0 for b, and -169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 9\left(-169\right)}}{2\times 9}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 9\left(-169\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{0±\sqrt{-36\left(-169\right)}}{2\times 9}$$

Multiply -36 times -169.

$$x=\frac{0±\sqrt{6084}}{2\times 9}$$

Take the square root of 6084.

$$x=\frac{0±78}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{0±78}{18}$$

Now solve the equation x=\frac{0±78}{18} when ± is plus. Reduce the fraction \frac{78}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{13}{3}$$

Now solve the equation x=\frac{0±78}{18} when ± is minus. Reduce the fraction \frac{-78}{18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{13}{3}$$

The equation is now solved.

$$x=\frac{13}{3} x=-\frac{13}{3}$$