Solve 1sqrt{x+7}+sqrt{3x-2}=5

$$1 \sqrt { x + 7 } + \sqrt { 3 x - 2 } = 5$$

Solve for x

$$x=2$$

Solution Steps

Subtract \sqrt{3x-2} from both sides of the equation.

$$\sqrt{x+7}=5-\sqrt{3x-2}$$

Reorder the terms.

$$\sqrt{x+7}=-\sqrt{3x-2}+5$$

Square both sides of the equation.

$$\left(\sqrt{x+7}\right)^{2}=\left(-\sqrt{3x-2}+5\right)^{2}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x+7=\left(-\sqrt{3x-2}+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\sqrt{3x-2}+5\right)^{2}.

$$x+7=\left(\sqrt{3x-2}\right)^{2}-10\sqrt{3x-2}+25$$

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

$$x+7=3x-2-10\sqrt{3x-2}+25$$

Add -2 and 25 to get 23.

$$x+7=3x+23-10\sqrt{3x-2}$$

Subtract 3x+23 from both sides of the equation.

$$x+7-\left(3x+23\right)=-10\sqrt{3x-2}$$

To find the opposite of 3x+23, find the opposite of each term.

$$x+7-3x-23=-10\sqrt{3x-2}$$

Combine x and -3x to get -2x.

$$-2x+7-23=-10\sqrt{3x-2}$$

Subtract 23 from 7 to get -16.

$$-2x-16=-10\sqrt{3x-2}$$

Square both sides of the equation.

$$\left(-2x-16\right)^{2}=\left(-10\sqrt{3x-2}\right)^{2}$$

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

$$4x^{2}+64x+256=\left(-10\sqrt{3x-2}\right)^{2}$$

Expand \left(-10\sqrt{3x-2}\right)^{2}.

$$4x^{2}+64x+256=\left(-10\right)^{2}\left(\sqrt{3x-2}\right)^{2}$$

Calculate -10 to the power of 2 and get 100.

$$4x^{2}+64x+256=100\left(\sqrt{3x-2}\right)^{2}$$

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

$$4x^{2}+64x+256=100\left(3x-2\right)$$

Use the distributive property to multiply 100 by 3x-2.

$$4x^{2}+64x+256=300x-200$$

Subtract 300x from both sides.

$$4x^{2}+64x+256-300x=-200$$

Combine 64x and -300x to get -236x.

$$4x^{2}-236x+256=-200$$

Add 200 to both sides.

$$4x^{2}-236x+256+200=0$$

Add 256 and 200 to get 456.

$$4x^{2}-236x+456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -236 for b, and 456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-236\right)±\sqrt{\left(-236\right)^{2}-4\times 4\times 456}}{2\times 4}$$

Square -236.

$$x=\frac{-\left(-236\right)±\sqrt{55696-4\times 4\times 456}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-236\right)±\sqrt{55696-16\times 456}}{2\times 4}$$

Multiply -16 times 456.

$$x=\frac{-\left(-236\right)±\sqrt{55696-7296}}{2\times 4}$$

Add 55696 to -7296.

$$x=\frac{-\left(-236\right)±\sqrt{48400}}{2\times 4}$$

Take the square root of 48400.

$$x=\frac{-\left(-236\right)±220}{2\times 4}$$

The opposite of -236 is 236.

$$x=\frac{236±220}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{236±220}{8}$$

Now solve the equation x=\frac{236±220}{8} when ± is plus. Add 236 to 220.

$$x=\frac{456}{8}$$

Divide 456 by 8.

$$x=57$$

Now solve the equation x=\frac{236±220}{8} when ± is minus. Subtract 220 from 236.

$$x=\frac{16}{8}$$

Divide 16 by 8.

$$x=2$$

The equation is now solved.

$$x=57 x=2$$

Substitute 57 for x in the equation 1\sqrt{x+7}+\sqrt{3x-2}=5.

$$1\sqrt{57+7}+\sqrt{3\times 57-2}=5$$

Simplify. The value x=57 does not satisfy the equation.

$$21=5$$

Substitute 2 for x in the equation 1\sqrt{x+7}+\sqrt{3x-2}=5.

$$1\sqrt{2+7}+\sqrt{3\times 2-2}=5$$

Simplify. The value x=2 satisfies the equation.

$$5=5$$

Equation \sqrt{x+7}=-\sqrt{3x-2}+5 has a unique solution.

$$x=2$$