Solve sqrt{5x-4}+sqrt{2x+1}=7

$$\sqrt{ 5x-4 } + \sqrt{ 2x+1 } =7$$

Solve for x

$$x=4$$

Solution Steps

Subtract \sqrt{2x+1} from both sides of the equation.

$$\sqrt{5x-4}=7-\sqrt{2x+1}$$

Square both sides of the equation.

$$\left(\sqrt{5x-4}\right)^{2}=\left(7-\sqrt{2x+1}\right)^{2}$$

Calculate \sqrt{5x-4} to the power of 2 and get 5x-4.

$$5x-4=\left(7-\sqrt{2x+1}\right)^{2}$$

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

$$5x-4=49-14\sqrt{2x+1}+\left(\sqrt{2x+1}\right)^{2}$$

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

$$5x-4=49-14\sqrt{2x+1}+2x+1$$

Add 49 and 1 to get 50.

$$5x-4=50-14\sqrt{2x+1}+2x$$

Subtract 50+2x from both sides of the equation.

$$5x-4-\left(50+2x\right)=-14\sqrt{2x+1}$$

To find the opposite of 50+2x, find the opposite of each term.

$$5x-4-50-2x=-14\sqrt{2x+1}$$

Subtract 50 from -4 to get -54.

$$5x-54-2x=-14\sqrt{2x+1}$$

Combine 5x and -2x to get 3x.

$$3x-54=-14\sqrt{2x+1}$$

Square both sides of the equation.

$$\left(3x-54\right)^{2}=\left(-14\sqrt{2x+1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-54\right)^{2}.

$$9x^{2}-324x+2916=\left(-14\sqrt{2x+1}\right)^{2}$$

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

$$9x^{2}-324x+2916=\left(-14\right)^{2}\left(\sqrt{2x+1}\right)^{2}$$

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

$$9x^{2}-324x+2916=196\left(\sqrt{2x+1}\right)^{2}$$

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

$$9x^{2}-324x+2916=196\left(2x+1\right)$$

Use the distributive property to multiply 196 by 2x+1.

$$9x^{2}-324x+2916=392x+196$$

Subtract 392x from both sides.

$$9x^{2}-324x+2916-392x=196$$

Combine -324x and -392x to get -716x.

$$9x^{2}-716x+2916=196$$

Subtract 196 from both sides.

$$9x^{2}-716x+2916-196=0$$

Subtract 196 from 2916 to get 2720.

$$9x^{2}-716x+2720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -716 for b, and 2720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-716\right)±\sqrt{\left(-716\right)^{2}-4\times 9\times 2720}}{2\times 9}$$

Square -716.

$$x=\frac{-\left(-716\right)±\sqrt{512656-4\times 9\times 2720}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-716\right)±\sqrt{512656-36\times 2720}}{2\times 9}$$

Multiply -36 times 2720.

$$x=\frac{-\left(-716\right)±\sqrt{512656-97920}}{2\times 9}$$

Add 512656 to -97920.

$$x=\frac{-\left(-716\right)±\sqrt{414736}}{2\times 9}$$

Take the square root of 414736.

$$x=\frac{-\left(-716\right)±644}{2\times 9}$$

The opposite of -716 is 716.

$$x=\frac{716±644}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{716±644}{18}$$

Now solve the equation x=\frac{716±644}{18} when ± is plus. Add 716 to 644.

$$x=\frac{1360}{18}$$

Reduce the fraction \frac{1360}{18} to lowest terms by extracting and canceling out 2.

$$x=\frac{680}{9}$$

Now solve the equation x=\frac{716±644}{18} when ± is minus. Subtract 644 from 716.

$$x=\frac{72}{18}$$

Divide 72 by 18.

$$x=4$$

The equation is now solved.

$$x=\frac{680}{9} x=4$$

Substitute \frac{680}{9} for x in the equation \sqrt{5x-4}+\sqrt{2x+1}=7.

$$\sqrt{5\times \frac{680}{9}-4}+\sqrt{2\times \frac{680}{9}+1}=7$$

Simplify. The value x=\frac{680}{9} does not satisfy the equation.

$$\frac{95}{3}=7$$

Substitute 4 for x in the equation \sqrt{5x-4}+\sqrt{2x+1}=7.

$$\sqrt{5\times 4-4}+\sqrt{2\times 4+1}=7$$

Simplify. The value x=4 satisfies the equation.

$$7=7$$

Equation \sqrt{5x-4}=-\sqrt{2x+1}+7 has a unique solution.

$$x=4$$