Solve sqrt{6x-1}-sqrt{5x+4}=9

$$\sqrt { 6 x - 1 } - \sqrt { 5 x + 4 } = 9$$

Solve for x

$$x=18\sqrt{2459}+896\approx 1788.589491312$$

Solution Steps

Subtract -\sqrt{5x+4} from both sides of the equation.

$$\sqrt{6x-1}=9+\sqrt{5x+4}$$

Square both sides of the equation.

$$\left(\sqrt{6x-1}\right)^{2}=\left(9+\sqrt{5x+4}\right)^{2}$$

Calculate \sqrt{6x-1} to the power of 2 and get 6x-1.

$$6x-1=\left(9+\sqrt{5x+4}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9+\sqrt{5x+4}\right)^{2}.

$$6x-1=81+18\sqrt{5x+4}+\left(\sqrt{5x+4}\right)^{2}$$

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

$$6x-1=81+18\sqrt{5x+4}+5x+4$$

Add 81 and 4 to get 85.

$$6x-1=85+18\sqrt{5x+4}+5x$$

Subtract 85+5x from both sides of the equation.

$$6x-1-\left(85+5x\right)=18\sqrt{5x+4}$$

To find the opposite of 85+5x, find the opposite of each term.

$$6x-1-85-5x=18\sqrt{5x+4}$$

Subtract 85 from -1 to get -86.

$$6x-86-5x=18\sqrt{5x+4}$$

Combine 6x and -5x to get x.

$$x-86=18\sqrt{5x+4}$$

Square both sides of the equation.

$$\left(x-86\right)^{2}=\left(18\sqrt{5x+4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-86\right)^{2}.

$$x^{2}-172x+7396=\left(18\sqrt{5x+4}\right)^{2}$$

Expand \left(18\sqrt{5x+4}\right)^{2}.

$$x^{2}-172x+7396=18^{2}\left(\sqrt{5x+4}\right)^{2}$$

Calculate 18 to the power of 2 and get 324.

$$x^{2}-172x+7396=324\left(\sqrt{5x+4}\right)^{2}$$

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

$$x^{2}-172x+7396=324\left(5x+4\right)$$

Use the distributive property to multiply 324 by 5x+4.

$$x^{2}-172x+7396=1620x+1296$$

Subtract 1620x from both sides.

$$x^{2}-172x+7396-1620x=1296$$

Combine -172x and -1620x to get -1792x.

$$x^{2}-1792x+7396=1296$$

Subtract 1296 from both sides.

$$x^{2}-1792x+7396-1296=0$$

Subtract 1296 from 7396 to get 6100.

$$x^{2}-1792x+6100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1792 for b, and 6100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1792\right)±\sqrt{\left(-1792\right)^{2}-4\times 6100}}{2}$$

Square -1792.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-4\times 6100}}{2}$$

Multiply -4 times 6100.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-24400}}{2}$$

Add 3211264 to -24400.

$$x=\frac{-\left(-1792\right)±\sqrt{3186864}}{2}$$

Take the square root of 3186864.

$$x=\frac{-\left(-1792\right)±36\sqrt{2459}}{2}$$

The opposite of -1792 is 1792.

$$x=\frac{1792±36\sqrt{2459}}{2}$$

Now solve the equation x=\frac{1792±36\sqrt{2459}}{2} when ± is plus. Add 1792 to 36\sqrt{2459}.

$$x=\frac{36\sqrt{2459}+1792}{2}$$

Divide 1792+36\sqrt{2459} by 2.

$$x=18\sqrt{2459}+896$$

Now solve the equation x=\frac{1792±36\sqrt{2459}}{2} when ± is minus. Subtract 36\sqrt{2459} from 1792.

$$x=\frac{1792-36\sqrt{2459}}{2}$$

Divide 1792-36\sqrt{2459} by 2.

$$x=896-18\sqrt{2459}$$

The equation is now solved.

$$x=18\sqrt{2459}+896 x=896-18\sqrt{2459}$$

Substitute 18\sqrt{2459}+896 for x in the equation \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Simplify. The value x=18\sqrt{2459}+896 satisfies the equation.

$$9=9$$

Substitute 896-18\sqrt{2459} for x in the equation \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(896-18\sqrt{2459}\right)-1}-\sqrt{5\left(896-18\sqrt{2459}\right)+4}=9$$

Simplify. The value x=896-18\sqrt{2459} does not satisfy the equation because the left and the right hand side have opposite signs.

$$99-2\times 2459^{\frac{1}{2}}=9$$

Substitute 18\sqrt{2459}+896 for x in the equation \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Simplify. The value x=18\sqrt{2459}+896 satisfies the equation.

$$9=9$$

Equation \sqrt{6x-1}=\sqrt{5x+4}+9 has a unique solution.

$$x=18\sqrt{2459}+896$$