Solve 25x^2=16x^2+{left(108/5right)}^2

$$25 { x }^{ 2 } =16 { x }^{ 2 } + { \left( \frac{ 108 }{ 5 } \right) }^{ 2 }$$

Solve for x

$$x = \frac{36}{5} = 7\frac{1}{5} = 7.2$$

$$x = -\frac{36}{5} = -7\frac{1}{5} = -7.2$$

Steps Using Difference of Squares

Calculate \frac{108}{5} to the power of 2 and get \frac{11664}{25}.

$$25x^{2}=16x^{2}+\frac{11664}{25}$$

Subtract 16x^{2} from both sides.

$$25x^{2}-16x^{2}=\frac{11664}{25}$$

Combine 25x^{2} and -16x^{2} to get 9x^{2}.

$$9x^{2}=\frac{11664}{25}$$

Subtract \frac{11664}{25} from both sides.

$$9x^{2}-\frac{11664}{25}=0$$

Divide both sides by \frac{9}{25}.

$$25x^{2}-1296=0$$

Consider 25x^{2}-1296. Rewrite 25x^{2}-1296 as \left(5x\right)^{2}-36^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(5x-36\right)\left(5x+36\right)=0$$

To find equation solutions, solve 5x-36=0 and 5x+36=0.

$$x=\frac{36}{5} x=-\frac{36}{5}$$

Steps by Finding Square Root

Calculate \frac{108}{5} to the power of 2 and get \frac{11664}{25}.

$$25x^{2}=16x^{2}+\frac{11664}{25}$$

Subtract 16x^{2} from both sides.

$$25x^{2}-16x^{2}=\frac{11664}{25}$$

Combine 25x^{2} and -16x^{2} to get 9x^{2}.

$$9x^{2}=\frac{11664}{25}$$

Divide both sides by 9.

$$x^{2}=\frac{\frac{11664}{25}}{9}$$

Express \frac{\frac{11664}{25}}{9} as a single fraction.

$$x^{2}=\frac{11664}{25\times 9}$$

Multiply 25 and 9 to get 225.

$$x^{2}=\frac{11664}{225}$$

Reduce the fraction \frac{11664}{225} to lowest terms by extracting and canceling out 9.

$$x^{2}=\frac{1296}{25}$$

Take the square root of both sides of the equation.

$$x=\frac{36}{5} x=-\frac{36}{5}$$

Steps Using the Quadratic Formula

Calculate \frac{108}{5} to the power of 2 and get \frac{11664}{25}.

$$25x^{2}=16x^{2}+\frac{11664}{25}$$

Subtract 16x^{2} from both sides.

$$25x^{2}-16x^{2}=\frac{11664}{25}$$

Combine 25x^{2} and -16x^{2} to get 9x^{2}.

$$9x^{2}=\frac{11664}{25}$$

Subtract \frac{11664}{25} from both sides.

$$9x^{2}-\frac{11664}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 0 for b, and -\frac{11664}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 9\left(-\frac{11664}{25}\right)}}{2\times 9}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 9\left(-\frac{11664}{25}\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{0±\sqrt{-36\left(-\frac{11664}{25}\right)}}{2\times 9}$$

Multiply -36 times -\frac{11664}{25}.

$$x=\frac{0±\sqrt{\frac{419904}{25}}}{2\times 9}$$

Take the square root of \frac{419904}{25}.

$$x=\frac{0±\frac{648}{5}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{0±\frac{648}{5}}{18}$$

Now solve the equation x=\frac{0±\frac{648}{5}}{18} when ± is plus.

$$x=\frac{36}{5}$$

Now solve the equation x=\frac{0±\frac{648}{5}}{18} when ± is minus.

$$x=-\frac{36}{5}$$

The equation is now solved.

$$x=\frac{36}{5} x=-\frac{36}{5}$$