Solve left(5xright)^2+left(12xright)^2=12^2

$${ \left(5x \right) }^{ 2 } + { \left(12x \right) }^{ 2 } = { 12 }^{ 2 }$$

Solve for x

$$x=-\frac{12}{13}\approx -0.923076923$$

$$x=\frac{12}{13}\approx 0.923076923$$

Steps Using Difference of Squares

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+\left(12x\right)^{2}=12^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+\left(12x\right)^{2}=12^{2}$$

Expand \left(12x\right)^{2}.

$$25x^{2}+12^{2}x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$25x^{2}+144x^{2}=12^{2}$$

Combine 25x^{2} and 144x^{2} to get 169x^{2}.

$$169x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$169x^{2}=144$$

Subtract 144 from both sides.

$$169x^{2}-144=0$$

Consider 169x^{2}-144. Rewrite 169x^{2}-144 as \left(13x\right)^{2}-12^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(13x-12\right)\left(13x+12\right)=0$$

To find equation solutions, solve 13x-12=0 and 13x+12=0.

$$x=\frac{12}{13} x=-\frac{12}{13}$$

Steps by Finding Square Root

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+\left(12x\right)^{2}=12^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+\left(12x\right)^{2}=12^{2}$$

Expand \left(12x\right)^{2}.

$$25x^{2}+12^{2}x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$25x^{2}+144x^{2}=12^{2}$$

Combine 25x^{2} and 144x^{2} to get 169x^{2}.

$$169x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$169x^{2}=144$$

Divide both sides by 169.

$$x^{2}=\frac{144}{169}$$

Take the square root of both sides of the equation.

$$x=\frac{12}{13} x=-\frac{12}{13}$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+\left(12x\right)^{2}=12^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+\left(12x\right)^{2}=12^{2}$$

Expand \left(12x\right)^{2}.

$$25x^{2}+12^{2}x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$25x^{2}+144x^{2}=12^{2}$$

Combine 25x^{2} and 144x^{2} to get 169x^{2}.

$$169x^{2}=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$169x^{2}=144$$

Subtract 144 from both sides.

$$169x^{2}-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 169 for a, 0 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 169\left(-144\right)}}{2\times 169}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 169\left(-144\right)}}{2\times 169}$$

Multiply -4 times 169.

$$x=\frac{0±\sqrt{-676\left(-144\right)}}{2\times 169}$$

Multiply -676 times -144.

$$x=\frac{0±\sqrt{97344}}{2\times 169}$$

Take the square root of 97344.

$$x=\frac{0±312}{2\times 169}$$

Multiply 2 times 169.

$$x=\frac{0±312}{338}$$

Now solve the equation x=\frac{0±312}{338} when ± is plus. Reduce the fraction \frac{312}{338} to lowest terms by extracting and canceling out 26.

$$x=\frac{12}{13}$$

Now solve the equation x=\frac{0±312}{338} when ± is minus. Reduce the fraction \frac{-312}{338} to lowest terms by extracting and canceling out 26.

$$x=-\frac{12}{13}$$

The equation is now solved.

$$x=\frac{12}{13} x=-\frac{12}{13}$$