Solve sqrt{2x+5}+sqrt{5x-6}=5

$$\sqrt{ 2x+5 } + \sqrt{ 5x-6 } =5$$

Solve for x

$$x=2$$

Solution Steps

Subtract \sqrt{5x-6} from both sides of the equation.

$$\sqrt{2x+5}=5-\sqrt{5x-6}$$

Square both sides of the equation.

$$\left(\sqrt{2x+5}\right)^{2}=\left(5-\sqrt{5x-6}\right)^{2}$$

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

$$2x+5=\left(5-\sqrt{5x-6}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-\sqrt{5x-6}\right)^{2}.

$$2x+5=25-10\sqrt{5x-6}+\left(\sqrt{5x-6}\right)^{2}$$

Calculate \sqrt{5x-6} to the power of 2 and get 5x-6.

$$2x+5=25-10\sqrt{5x-6}+5x-6$$

Subtract 6 from 25 to get 19.

$$2x+5=19-10\sqrt{5x-6}+5x$$

Subtract 19+5x from both sides of the equation.

$$2x+5-\left(19+5x\right)=-10\sqrt{5x-6}$$

To find the opposite of 19+5x, find the opposite of each term.

$$2x+5-19-5x=-10\sqrt{5x-6}$$

Subtract 19 from 5 to get -14.

$$2x-14-5x=-10\sqrt{5x-6}$$

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

$$-3x-14=-10\sqrt{5x-6}$$

Square both sides of the equation.

$$\left(-3x-14\right)^{2}=\left(-10\sqrt{5x-6}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-3x-14\right)^{2}.

$$9x^{2}+84x+196=\left(-10\sqrt{5x-6}\right)^{2}$$

Expand \left(-10\sqrt{5x-6}\right)^{2}.

$$9x^{2}+84x+196=\left(-10\right)^{2}\left(\sqrt{5x-6}\right)^{2}$$

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

$$9x^{2}+84x+196=100\left(\sqrt{5x-6}\right)^{2}$$

Calculate \sqrt{5x-6} to the power of 2 and get 5x-6.

$$9x^{2}+84x+196=100\left(5x-6\right)$$

Use the distributive property to multiply 100 by 5x-6.

$$9x^{2}+84x+196=500x-600$$

Subtract 500x from both sides.

$$9x^{2}+84x+196-500x=-600$$

Combine 84x and -500x to get -416x.

$$9x^{2}-416x+196=-600$$

Add 600 to both sides.

$$9x^{2}-416x+196+600=0$$

Add 196 and 600 to get 796.

$$9x^{2}-416x+796=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -416 for b, and 796 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-416\right)±\sqrt{\left(-416\right)^{2}-4\times 9\times 796}}{2\times 9}$$

Square -416.

$$x=\frac{-\left(-416\right)±\sqrt{173056-4\times 9\times 796}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-416\right)±\sqrt{173056-36\times 796}}{2\times 9}$$

Multiply -36 times 796.

$$x=\frac{-\left(-416\right)±\sqrt{173056-28656}}{2\times 9}$$

Add 173056 to -28656.

$$x=\frac{-\left(-416\right)±\sqrt{144400}}{2\times 9}$$

Take the square root of 144400.

$$x=\frac{-\left(-416\right)±380}{2\times 9}$$

The opposite of -416 is 416.

$$x=\frac{416±380}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{416±380}{18}$$

Now solve the equation x=\frac{416±380}{18} when ± is plus. Add 416 to 380.

$$x=\frac{796}{18}$$

Reduce the fraction \frac{796}{18} to lowest terms by extracting and canceling out 2.

$$x=\frac{398}{9}$$

Now solve the equation x=\frac{416±380}{18} when ± is minus. Subtract 380 from 416.

$$x=\frac{36}{18}$$

Divide 36 by 18.

$$x=2$$

The equation is now solved.

$$x=\frac{398}{9} x=2$$

Substitute \frac{398}{9} for x in the equation \sqrt{2x+5}+\sqrt{5x-6}=5.

$$\sqrt{2\times \frac{398}{9}+5}+\sqrt{5\times \frac{398}{9}-6}=5$$

Simplify. The value x=\frac{398}{9} does not satisfy the equation.

$$\frac{73}{3}=5$$

Substitute 2 for x in the equation \sqrt{2x+5}+\sqrt{5x-6}=5.

$$\sqrt{2\times 2+5}+\sqrt{5\times 2-6}=5$$

Simplify. The value x=2 satisfies the equation.

$$5=5$$

Equation \sqrt{2x+5}=-\sqrt{5x-6}+5 has a unique solution.

$$x=2$$