Solve sqrt{x-5}+3sqrt{x+3}=10

$$\sqrt { x - 5 } + 3 \sqrt { x + 3 } = 10$$

Solve for x

$$x=6$$

Solution Steps

Subtract 3\sqrt{x+3} from both sides of the equation.

$$\sqrt{x-5}=10-3\sqrt{x+3}$$

Square both sides of the equation.

$$\left(\sqrt{x-5}\right)^{2}=\left(10-3\sqrt{x+3}\right)^{2}$$

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$x-5=\left(10-3\sqrt{x+3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-3\sqrt{x+3}\right)^{2}.

$$x-5=100-60\sqrt{x+3}+9\left(\sqrt{x+3}\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$x-5=100-60\sqrt{x+3}+9\left(x+3\right)$$

Use the distributive property to multiply 9 by x+3.

$$x-5=100-60\sqrt{x+3}+9x+27$$

Add 100 and 27 to get 127.

$$x-5=127-60\sqrt{x+3}+9x$$

Subtract 127+9x from both sides of the equation.

$$x-5-\left(127+9x\right)=-60\sqrt{x+3}$$

To find the opposite of 127+9x, find the opposite of each term.

$$x-5-127-9x=-60\sqrt{x+3}$$

Subtract 127 from -5 to get -132.

$$x-132-9x=-60\sqrt{x+3}$$

Combine x and -9x to get -8x.

$$-8x-132=-60\sqrt{x+3}$$

Square both sides of the equation.

$$\left(-8x-132\right)^{2}=\left(-60\sqrt{x+3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-8x-132\right)^{2}.

$$64x^{2}+2112x+17424=\left(-60\sqrt{x+3}\right)^{2}$$

Expand \left(-60\sqrt{x+3}\right)^{2}.

$$64x^{2}+2112x+17424=\left(-60\right)^{2}\left(\sqrt{x+3}\right)^{2}$$

Calculate -60 to the power of 2 and get 3600.

$$64x^{2}+2112x+17424=3600\left(\sqrt{x+3}\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$64x^{2}+2112x+17424=3600\left(x+3\right)$$

Use the distributive property to multiply 3600 by x+3.

$$64x^{2}+2112x+17424=3600x+10800$$

Subtract 3600x from both sides.

$$64x^{2}+2112x+17424-3600x=10800$$

Combine 2112x and -3600x to get -1488x.

$$64x^{2}-1488x+17424=10800$$

Subtract 10800 from both sides.

$$64x^{2}-1488x+17424-10800=0$$

Subtract 10800 from 17424 to get 6624.

$$64x^{2}-1488x+6624=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, -1488 for b, and 6624 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1488\right)±\sqrt{\left(-1488\right)^{2}-4\times 64\times 6624}}{2\times 64}$$

Square -1488.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-4\times 64\times 6624}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-256\times 6624}}{2\times 64}$$

Multiply -256 times 6624.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-1695744}}{2\times 64}$$

Add 2214144 to -1695744.

$$x=\frac{-\left(-1488\right)±\sqrt{518400}}{2\times 64}$$

Take the square root of 518400.

$$x=\frac{-\left(-1488\right)±720}{2\times 64}$$

The opposite of -1488 is 1488.

$$x=\frac{1488±720}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{1488±720}{128}$$

Now solve the equation x=\frac{1488±720}{128} when ± is plus. Add 1488 to 720.

$$x=\frac{2208}{128}$$

Reduce the fraction \frac{2208}{128} to lowest terms by extracting and canceling out 32.

$$x=\frac{69}{4}$$

Now solve the equation x=\frac{1488±720}{128} when ± is minus. Subtract 720 from 1488.

$$x=\frac{768}{128}$$

Divide 768 by 128.

$$x=6$$

The equation is now solved.

$$x=\frac{69}{4} x=6$$

Substitute \frac{69}{4} for x in the equation \sqrt{x-5}+3\sqrt{x+3}=10.

$$\sqrt{\frac{69}{4}-5}+3\sqrt{\frac{69}{4}+3}=10$$

Simplify. The value x=\frac{69}{4} does not satisfy the equation.

$$17=10$$

Substitute 6 for x in the equation \sqrt{x-5}+3\sqrt{x+3}=10.

$$\sqrt{6-5}+3\sqrt{6+3}=10$$

Simplify. The value x=6 satisfies the equation.

$$10=10$$

Equation \sqrt{x-5}=-3\sqrt{x+3}+10 has a unique solution.

$$x=6$$