Solve x+sqrt{x^2+16}=40/sqrt{x^2+16}

$$x+ \sqrt{ { x }^{ 2 } +16 } = \frac{ 40 }{ \sqrt{ { x }^{ 2 } +16 } }$$

Solve for x

$$x=3$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x^{2}+16}=\frac{40}{\sqrt{x^{2}+16}}-x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}}.

$$\sqrt{x^{2}+16}=\frac{40}{\sqrt{x^{2}+16}}-\frac{x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}}$$

Since \frac{40}{\sqrt{x^{2}+16}} and \frac{x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{x^{2}+16}=\frac{40-x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}+16}\right)^{2}=\left(\frac{40-x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}}\right)^{2}$$

Calculate \sqrt{x^{2}+16} to the power of 2 and get x^{2}+16.

$$x^{2}+16=\left(\frac{40-x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}}\right)^{2}$$

To raise \frac{40-x\sqrt{x^{2}+16}}{\sqrt{x^{2}+16}} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+16=\frac{\left(40-x\sqrt{x^{2}+16}\right)^{2}}{\left(\sqrt{x^{2}+16}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(40-x\sqrt{x^{2}+16}\right)^{2}.

$$x^{2}+16=\frac{1600-80x\sqrt{x^{2}+16}+x^{2}\left(\sqrt{x^{2}+16}\right)^{2}}{\left(\sqrt{x^{2}+16}\right)^{2}}$$

Calculate \sqrt{x^{2}+16} to the power of 2 and get x^{2}+16.

$$x^{2}+16=\frac{1600-80x\sqrt{x^{2}+16}+x^{2}\left(x^{2}+16\right)}{\left(\sqrt{x^{2}+16}\right)^{2}}$$

Use the distributive property to multiply x^{2} by x^{2}+16.

$$x^{2}+16=\frac{1600-80x\sqrt{x^{2}+16}+x^{4}+16x^{2}}{\left(\sqrt{x^{2}+16}\right)^{2}}$$

Calculate \sqrt{x^{2}+16} to the power of 2 and get x^{2}+16.

$$x^{2}+16=\frac{1600-80x\sqrt{x^{2}+16}+x^{4}+16x^{2}}{x^{2}+16}$$

Multiply both sides of the equation by x^{2}+16.

$$\left(x^{2}+16\right)x^{2}+\left(x^{2}+16\right)\times 16=1600-80x\sqrt{x^{2}+16}+x^{4}+16x^{2}$$

Subtract 1600+x^{4}+16x^{2} from both sides of the equation.

$$\left(x^{2}+16\right)x^{2}+\left(x^{2}+16\right)\times 16-\left(1600+x^{4}+16x^{2}\right)=-80x\sqrt{x^{2}+16}$$

Use the distributive property to multiply x^{2}+16 by x^{2}.

$$x^{4}+16x^{2}+\left(x^{2}+16\right)\times 16-\left(1600+x^{4}+16x^{2}\right)=-80x\sqrt{x^{2}+16}$$

Use the distributive property to multiply x^{2}+16 by 16.

$$x^{4}+16x^{2}+16x^{2}+256-\left(1600+x^{4}+16x^{2}\right)=-80x\sqrt{x^{2}+16}$$

Combine 16x^{2} and 16x^{2} to get 32x^{2}.

$$x^{4}+32x^{2}+256-\left(1600+x^{4}+16x^{2}\right)=-80x\sqrt{x^{2}+16}$$

To find the opposite of 1600+x^{4}+16x^{2}, find the opposite of each term.

$$x^{4}+32x^{2}+256-1600-x^{4}-16x^{2}=-80x\sqrt{x^{2}+16}$$

Subtract 1600 from 256 to get -1344.

$$x^{4}+32x^{2}-1344-x^{4}-16x^{2}=-80x\sqrt{x^{2}+16}$$

Combine x^{4} and -x^{4} to get 0.

$$32x^{2}-1344-16x^{2}=-80x\sqrt{x^{2}+16}$$

Combine 32x^{2} and -16x^{2} to get 16x^{2}.

$$16x^{2}-1344=-80x\sqrt{x^{2}+16}$$

Square both sides of the equation.

$$\left(16x^{2}-1344\right)^{2}=\left(-80x\sqrt{x^{2}+16}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16x^{2}-1344\right)^{2}.

$$256\left(x^{2}\right)^{2}-43008x^{2}+1806336=\left(-80x\sqrt{x^{2}+16}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$256x^{4}-43008x^{2}+1806336=\left(-80x\sqrt{x^{2}+16}\right)^{2}$$

Expand \left(-80x\sqrt{x^{2}+16}\right)^{2}.

$$256x^{4}-43008x^{2}+1806336=\left(-80\right)^{2}x^{2}\left(\sqrt{x^{2}+16}\right)^{2}$$

Calculate -80 to the power of 2 and get 6400.

$$256x^{4}-43008x^{2}+1806336=6400x^{2}\left(\sqrt{x^{2}+16}\right)^{2}$$

Calculate \sqrt{x^{2}+16} to the power of 2 and get x^{2}+16.

$$256x^{4}-43008x^{2}+1806336=6400x^{2}\left(x^{2}+16\right)$$

Use the distributive property to multiply 6400x^{2} by x^{2}+16.

$$256x^{4}-43008x^{2}+1806336=6400x^{4}+102400x^{2}$$

Subtract 6400x^{4} from both sides.

$$256x^{4}-43008x^{2}[phone]x^{4}=102400x^{2}$$

Combine 256x^{4} and -6400x^{4} to get -6144x^{4}.

$$-6144x^{4}-43008x^{2}+1806336=102400x^{2}$$

Subtract 102400x^{2} from both sides.

$$-6144x^{4}-43008x^{2}[phone]x^{2}=0$$

Combine -43008x^{2} and -102400x^{2} to get -145408x^{2}.

$$-6144x^{4}-145408x^{2}+1806336=0$$

Substitute t for x^{2}.

$$-6144t^{2}-145408t+1806336=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -6144 for a, -145408 for b, and 1806336 for c in the quadratic formula.

$$t=\frac{-\left(-145408\right)±\sqrt{\left(-145408\right)^{2}-4\left(-6144\right)\times 1806336}}{-6144\times 2}$$

Do the calculations.

$$t=\frac{145408±256000}{-12288}$$

Solve the equation t=\frac{145408±256000}{-12288} when ± is plus and when ± is minus.

$$t=-\frac{98}{3} t=9$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=3 x=-3$$

Substitute 3 for x in the equation x+\sqrt{x^{2}+16}=\frac{40}{\sqrt{x^{2}+16}}.

$$3+\sqrt{3^{2}+16}=\frac{40}{\sqrt{3^{2}+16}}$$

Simplify. The value x=3 satisfies the equation.

$$8=8$$

Substitute -3 for x in the equation x+\sqrt{x^{2}+16}=\frac{40}{\sqrt{x^{2}+16}}.

$$-3+\sqrt{\left(-3\right)^{2}+16}=\frac{40}{\sqrt{\left(-3\right)^{2}+16}}$$

Simplify. The value x=-3 does not satisfy the equation.

$$2=8$$

Substitute 3 for x in the equation x+\sqrt{x^{2}+16}=\frac{40}{\sqrt{x^{2}+16}}.

$$3+\sqrt{3^{2}+16}=\frac{40}{\sqrt{3^{2}+16}}$$

Simplify. The value x=3 satisfies the equation.

$$8=8$$

Equation \sqrt{x^{2}+16}=\frac{-x\sqrt{x^{2}+16}+40}{\sqrt{x^{2}+16}} has a unique solution.

$$x=3$$