Solve x^2+{left(-3/4xright)}^2=10^2

$${ x }^{ 2 } + { \left(- \frac{ 3 }{ 4 } x \right) }^{ 2 } = { 10 }^{ 2 }$$

Solve for x

$$x=-8$$

$$x=8$$

Steps Using Difference of Squares

Expand \left(-\frac{3}{4}x\right)^{2}.

$$x^{2}+\left(-\frac{3}{4}\right)^{2}x^{2}=10^{2}$$

Calculate -\frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$x^{2}+\frac{9}{16}x^{2}=10^{2}$$

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

$$\frac{25}{16}x^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{25}{16}x^{2}=100$$

Subtract 100 from both sides.

$$\frac{25}{16}x^{2}-100=0$$

Divide both sides by \frac{25}{16}.

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

Consider x^{2}-64. Rewrite x^{2}-64 as x^{2}-8^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-8\right)\left(x+8\right)=0$$

To find equation solutions, solve x-8=0 and x+8=0.

$$x=8 x=-8$$

Steps by Finding Square Root

Expand \left(-\frac{3}{4}x\right)^{2}.

$$x^{2}+\left(-\frac{3}{4}\right)^{2}x^{2}=10^{2}$$

Calculate -\frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$x^{2}+\frac{9}{16}x^{2}=10^{2}$$

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

$$\frac{25}{16}x^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{25}{16}x^{2}=100$$

Multiply both sides by \frac{16}{25}, the reciprocal of \frac{25}{16}.

$$x^{2}=100\times \frac{16}{25}$$

Multiply 100 and \frac{16}{25} to get 64.

$$x^{2}=64$$

Take the square root of both sides of the equation.

$$x=8 x=-8$$

Steps Using the Quadratic Formula

Expand \left(-\frac{3}{4}x\right)^{2}.

$$x^{2}+\left(-\frac{3}{4}\right)^{2}x^{2}=10^{2}$$

Calculate -\frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$x^{2}+\frac{9}{16}x^{2}=10^{2}$$

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

$$\frac{25}{16}x^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{25}{16}x^{2}=100$$

Subtract 100 from both sides.

$$\frac{25}{16}x^{2}-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{16} for a, 0 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{25}{16}\left(-100\right)}}{2\times \frac{25}{16}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{25}{16}\left(-100\right)}}{2\times \frac{25}{16}}$$

Multiply -4 times \frac{25}{16}.

$$x=\frac{0±\sqrt{-\frac{25}{4}\left(-100\right)}}{2\times \frac{25}{16}}$$

Multiply -\frac{25}{4} times -100.

$$x=\frac{0±\sqrt{625}}{2\times \frac{25}{16}}$$

Take the square root of 625.

$$x=\frac{0±25}{2\times \frac{25}{16}}$$

Multiply 2 times \frac{25}{16}.

$$x=\frac{0±25}{\frac{25}{8}}$$

Now solve the equation x=\frac{0±25}{\frac{25}{8}} when ± is plus. Divide 25 by \frac{25}{8} by multiplying 25 by the reciprocal of \frac{25}{8}.

$$x=8$$

Now solve the equation x=\frac{0±25}{\frac{25}{8}} when ± is minus. Divide -25 by \frac{25}{8} by multiplying -25 by the reciprocal of \frac{25}{8}.

$$x=-8$$

The equation is now solved.

$$x=8 x=-8$$