Solve {left(17x/15right)}^2-x^2=16

$${ \left(17 \frac{ x }{ 15 } \right) }^{ 2 } - { x }^{ 2 } =16$$

Solve for x

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

Steps Using Difference of Squares

Express 17\times \frac{x}{15} as a single fraction.

$$\left(\frac{17x}{15}\right)^{2}-x^{2}=16$$

To raise \frac{17x}{15} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(17x\right)^{2}}{15^{2}}-x^{2}=16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{15^{2}}{15^{2}}.

$$\frac{\left(17x\right)^{2}}{15^{2}}-\frac{x^{2}\times 15^{2}}{15^{2}}=16$$

Since \frac{\left(17x\right)^{2}}{15^{2}} and \frac{x^{2}\times 15^{2}}{15^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(17x\right)^{2}-x^{2}\times 15^{2}}{15^{2}}=16$$

Do the multiplications in \left(17x\right)^{2}-x^{2}\times 15^{2}.

$$\frac{\left(17x\right)^{2}-225x^{2}}{15^{2}}=16$$

Combine like terms in \left(17x\right)^{2}-225x^{2}.

$$\frac{64x^{2}}{15^{2}}=16$$

Calculate 15 to the power of 2 and get 225.

$$\frac{64x^{2}}{225}=16$$

Subtract 16 from both sides.

$$\frac{64x^{2}}{225}-16=0$$

Multiply both sides of the equation by 225.

$$64x^{2}-3600=0$$

Divide both sides by 16.

$$4x^{2}-225=0$$

Consider 4x^{2}-225. Rewrite 4x^{2}-225 as \left(2x\right)^{2}-15^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-15\right)\left(2x+15\right)=0$$

To find equation solutions, solve 2x-15=0 and 2x+15=0.

$$x=\frac{15}{2} x=-\frac{15}{2}$$

Steps by Finding Square Root

Express 17\times \frac{x}{15} as a single fraction.

$$\left(\frac{17x}{15}\right)^{2}-x^{2}=16$$

To raise \frac{17x}{15} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(17x\right)^{2}}{15^{2}}-x^{2}=16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{15^{2}}{15^{2}}.

$$\frac{\left(17x\right)^{2}}{15^{2}}-\frac{x^{2}\times 15^{2}}{15^{2}}=16$$

Since \frac{\left(17x\right)^{2}}{15^{2}} and \frac{x^{2}\times 15^{2}}{15^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(17x\right)^{2}-x^{2}\times 15^{2}}{15^{2}}=16$$

Do the multiplications in \left(17x\right)^{2}-x^{2}\times 15^{2}.

$$\frac{\left(17x\right)^{2}-225x^{2}}{15^{2}}=16$$

Combine like terms in \left(17x\right)^{2}-225x^{2}.

$$\frac{64x^{2}}{15^{2}}=16$$

Calculate 15 to the power of 2 and get 225.

$$\frac{64x^{2}}{225}=16$$

Multiply both sides by 225.

$$64x^{2}=16\times 225$$

Multiply 16 and 225 to get 3600.

$$64x^{2}=3600$$

Divide both sides by 64.

$$x^{2}=\frac{3600}{64}$$

Reduce the fraction \frac{3600}{64} to lowest terms by extracting and canceling out 16.

$$x^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$x=\frac{15}{2} x=-\frac{15}{2}$$

Steps Using the Quadratic Formula

Express 17\times \frac{x}{15} as a single fraction.

$$\left(\frac{17x}{15}\right)^{2}-x^{2}=16$$

To raise \frac{17x}{15} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(17x\right)^{2}}{15^{2}}-x^{2}=16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{15^{2}}{15^{2}}.

$$\frac{\left(17x\right)^{2}}{15^{2}}-\frac{x^{2}\times 15^{2}}{15^{2}}=16$$

Since \frac{\left(17x\right)^{2}}{15^{2}} and \frac{x^{2}\times 15^{2}}{15^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(17x\right)^{2}-x^{2}\times 15^{2}}{15^{2}}=16$$

Do the multiplications in \left(17x\right)^{2}-x^{2}\times 15^{2}.

$$\frac{\left(17x\right)^{2}-225x^{2}}{15^{2}}=16$$

Combine like terms in \left(17x\right)^{2}-225x^{2}.

$$\frac{64x^{2}}{15^{2}}=16$$

Calculate 15 to the power of 2 and get 225.

$$\frac{64x^{2}}{225}=16$$

Subtract 16 from both sides.

$$\frac{64x^{2}}{225}-16=0$$

Multiply both sides of the equation by 225.

$$64x^{2}-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, 0 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 64\left(-3600\right)}}{2\times 64}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 64\left(-3600\right)}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{0±\sqrt{-256\left(-3600\right)}}{2\times 64}$$

Multiply -256 times -3600.

$$x=\frac{0±\sqrt{921600}}{2\times 64}$$

Take the square root of 921600.

$$x=\frac{0±960}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{0±960}{128}$$

Now solve the equation x=\frac{0±960}{128} when ± is plus. Reduce the fraction \frac{960}{128} to lowest terms by extracting and canceling out 64.

$$x=\frac{15}{2}$$

Now solve the equation x=\frac{0±960}{128} when ± is minus. Reduce the fraction \frac{-960}{128} to lowest terms by extracting and canceling out 64.

$$x=-\frac{15}{2}$$

The equation is now solved.

$$x=\frac{15}{2} x=-\frac{15}{2}$$