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

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

Solve for x

$$x=5$$

Solution Steps

Subtract \sqrt{3x+1} from both sides of the equation.

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

Square both sides of the equation.

$$\left(\sqrt{2x-1}\right)^{2}=\left(7-\sqrt{3x+1}\right)^{2}$$

Calculate \sqrt{2x-1} to the power of 2 and get 2x-1.

$$2x-1=\left(7-\sqrt{3x+1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-\sqrt{3x+1}\right)^{2}.

$$2x-1=49-14\sqrt{3x+1}+\left(\sqrt{3x+1}\right)^{2}$$

Calculate \sqrt{3x+1} to the power of 2 and get 3x+1.

$$2x-1=49-14\sqrt{3x+1}+3x+1$$

Add 49 and 1 to get 50.

$$2x-1=50-14\sqrt{3x+1}+3x$$

Subtract 50+3x from both sides of the equation.

$$2x-1-\left(50+3x\right)=-14\sqrt{3x+1}$$

To find the opposite of 50+3x, find the opposite of each term.

$$2x-1-50-3x=-14\sqrt{3x+1}$$

Subtract 50 from -1 to get -51.

$$2x-51-3x=-14\sqrt{3x+1}$$

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

$$-x-51=-14\sqrt{3x+1}$$

Square both sides of the equation.

$$\left(-x-51\right)^{2}=\left(-14\sqrt{3x+1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-51\right)^{2}.

$$x^{2}+102x+2601=\left(-14\sqrt{3x+1}\right)^{2}$$

Expand \left(-14\sqrt{3x+1}\right)^{2}.

$$x^{2}+102x+2601=\left(-14\right)^{2}\left(\sqrt{3x+1}\right)^{2}$$

Calculate -14 to the power of 2 and get 196.

$$x^{2}+102x+2601=196\left(\sqrt{3x+1}\right)^{2}$$

Calculate \sqrt{3x+1} to the power of 2 and get 3x+1.

$$x^{2}+102x+2601=196\left(3x+1\right)$$

Use the distributive property to multiply 196 by 3x+1.

$$x^{2}+102x+2601=588x+196$$

Subtract 588x from both sides.

$$x^{2}+102x+2601-588x=196$$

Combine 102x and -588x to get -486x.

$$x^{2}-486x+2601=196$$

Subtract 196 from both sides.

$$x^{2}-486x+2601-196=0$$

Subtract 196 from 2601 to get 2405.

$$x^{2}-486x+2405=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -486 for b, and 2405 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-486\right)±\sqrt{\left(-486\right)^{2}-4\times 2405}}{2}$$

Square -486.

$$x=\frac{-\left(-486\right)±\sqrt{236196-4\times 2405}}{2}$$

Multiply -4 times 2405.

$$x=\frac{-\left(-486\right)±\sqrt{236196-9620}}{2}$$

Add 236196 to -9620.

$$x=\frac{-\left(-486\right)±\sqrt{226576}}{2}$$

Take the square root of 226576.

$$x=\frac{-\left(-486\right)±476}{2}$$

The opposite of -486 is 486.

$$x=\frac{486±476}{2}$$

Now solve the equation x=\frac{486±476}{2} when ± is plus. Add 486 to 476.

$$x=\frac{962}{2}$$

Divide 962 by 2.

$$x=481$$

Now solve the equation x=\frac{486±476}{2} when ± is minus. Subtract 476 from 486.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

The equation is now solved.

$$x=481 x=5$$

Substitute 481 for x in the equation \sqrt{2x-1}+\sqrt{3x+1}=7.

$$\sqrt{2\times 481-1}+\sqrt{3\times 481+1}=7$$

Simplify. The value x=481 does not satisfy the equation.

$$69=7$$

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

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

Simplify. The value x=5 satisfies the equation.

$$7=7$$

Equation \sqrt{2x-1}=-\sqrt{3x+1}+7 has a unique solution.

$$x=5$$