Solve 25^2/75^2+x^2/45^2=1

$$\frac{ { 25 }^{ 2 } }{ { 75 }^{ 2 } } + \frac{ { x }^{ 2 } }{ { 45 }^{ 2 } } = 1$$

Solve for x

$$x=30\sqrt{2}\approx 42.426406871$$

$$x=-30\sqrt{2}\approx -42.426406871$$

Steps by Finding Square Root

Calculate 25 to the power of 2 and get 625.

$$\frac{625}{75^{2}}+\frac{x^{2}}{45^{2}}=1$$

Calculate 75 to the power of 2 and get 5625.

$$\frac{625}{5625}+\frac{x^{2}}{45^{2}}=1$$

Reduce the fraction \frac{625}{5625} to lowest terms by extracting and canceling out 625.

$$\frac{1}{9}+\frac{x^{2}}{45^{2}}=1$$

Calculate 45 to the power of 2 and get 2025.

$$\frac{1}{9}+\frac{x^{2}}{2025}=1$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 2025 is 2025. Multiply \frac{1}{9} times \frac{225}{225}.

$$\frac{225}{2025}+\frac{x^{2}}{2025}=1$$

Since \frac{225}{2025} and \frac{x^{2}}{2025} have the same denominator, add them by adding their numerators.

$$\frac{225+x^{2}}{2025}=1$$

Divide each term of 225+x^{2} by 2025 to get \frac{1}{9}+\frac{1}{2025}x^{2}.

$$\frac{1}{9}+\frac{1}{2025}x^{2}=1$$

Subtract \frac{1}{9} from both sides.

$$\frac{1}{2025}x^{2}=1-\frac{1}{9}$$

Subtract \frac{1}{9} from 1 to get \frac{8}{9}.

$$\frac{1}{2025}x^{2}=\frac{8}{9}$$

Multiply both sides by 2025, the reciprocal of \frac{1}{2025}.

$$x^{2}=\frac{8}{9}\times 2025$$

Multiply \frac{8}{9} and 2025 to get 1800.

$$x^{2}=1800$$

Take the square root of both sides of the equation.

$$x=30\sqrt{2} x=-30\sqrt{2}$$

Steps Using the Quadratic Formula

Calculate 25 to the power of 2 and get 625.

$$\frac{625}{75^{2}}+\frac{x^{2}}{45^{2}}=1$$

Calculate 75 to the power of 2 and get 5625.

$$\frac{625}{5625}+\frac{x^{2}}{45^{2}}=1$$

Reduce the fraction \frac{625}{5625} to lowest terms by extracting and canceling out 625.

$$\frac{1}{9}+\frac{x^{2}}{45^{2}}=1$$

Calculate 45 to the power of 2 and get 2025.

$$\frac{1}{9}+\frac{x^{2}}{2025}=1$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 2025 is 2025. Multiply \frac{1}{9} times \frac{225}{225}.

$$\frac{225}{2025}+\frac{x^{2}}{2025}=1$$

Since \frac{225}{2025} and \frac{x^{2}}{2025} have the same denominator, add them by adding their numerators.

$$\frac{225+x^{2}}{2025}=1$$

Divide each term of 225+x^{2} by 2025 to get \frac{1}{9}+\frac{1}{2025}x^{2}.

$$\frac{1}{9}+\frac{1}{2025}x^{2}=1$$

Subtract 1 from both sides.

$$\frac{1}{9}+\frac{1}{2025}x^{2}-1=0$$

Subtract 1 from \frac{1}{9} to get -\frac{8}{9}.

$$-\frac{8}{9}+\frac{1}{2025}x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\frac{1}{2025}x^{2}-\frac{8}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2025} for a, 0 for b, and -\frac{8}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2025}\left(-\frac{8}{9}\right)}}{2\times \frac{1}{2025}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2025}\left(-\frac{8}{9}\right)}}{2\times \frac{1}{2025}}$$

Multiply -4 times \frac{1}{2025}.

$$x=\frac{0±\sqrt{-\frac{4}{2025}\left(-\frac{8}{9}\right)}}{2\times \frac{1}{2025}}$$

Multiply -\frac{4}{2025} times -\frac{8}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0±\sqrt{\frac{32}{18225}}}{2\times \frac{1}{2025}}$$

Take the square root of \frac{32}{18225}.

$$x=\frac{0±\frac{4\sqrt{2}}{135}}{2\times \frac{1}{2025}}$$

Multiply 2 times \frac{1}{2025}.

$$x=\frac{0±\frac{4\sqrt{2}}{135}}{\frac{2}{2025}}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{2}}{135}}{\frac{2}{2025}} when ± is plus.

$$x=30\sqrt{2}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{2}}{135}}{\frac{2}{2025}} when ± is minus.

$$x=-30\sqrt{2}$$

The equation is now solved.

$$x=30\sqrt{2} x=-30\sqrt{2}$$