Solve sqrt{15-x}-sqrt{6x}=-3

$$\sqrt{ 15-x } - \sqrt{ 6x } =-3$$

Solve for x

$$x=6$$

Solution Steps

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

$$\sqrt{15-x}=-3+\sqrt{6x}$$

Square both sides of the equation.

$$\left(\sqrt{15-x}\right)^{2}=\left(-3+\sqrt{6x}\right)^{2}$$

Calculate \sqrt{15-x} to the power of 2 and get 15-x.

$$15-x=\left(-3+\sqrt{6x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3+\sqrt{6x}\right)^{2}.

$$15-x=9-6\sqrt{6x}+\left(\sqrt{6x}\right)^{2}$$

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

$$15-x=9-6\sqrt{6x}+6x$$

Subtract 9+6x from both sides of the equation.

$$15-x-\left(9+6x\right)=-6\sqrt{6x}$$

To find the opposite of 9+6x, find the opposite of each term.

$$15-x-9-6x=-6\sqrt{6x}$$

Subtract 9 from 15 to get 6.

$$6-x-6x=-6\sqrt{6x}$$

Combine -x and -6x to get -7x.

$$6-7x=-6\sqrt{6x}$$

Square both sides of the equation.

$$\left(6-7x\right)^{2}=\left(-6\sqrt{6x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-7x\right)^{2}.

$$36-84x+49x^{2}=\left(-6\sqrt{6x}\right)^{2}$$

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

$$36-84x+49x^{2}=\left(-6\right)^{2}\left(\sqrt{6x}\right)^{2}$$

Calculate -6 to the power of 2 and get 36.

$$36-84x+49x^{2}=36\left(\sqrt{6x}\right)^{2}$$

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

$$36-84x+49x^{2}=36\times 6x$$

Multiply 36 and 6 to get 216.

$$36-84x+49x^{2}=216x$$

Subtract 216x from both sides.

$$36-84x+49x^{2}-216x=0$$

Combine -84x and -216x to get -300x.

$$36-300x+49x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$49x^{2}-300x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, -300 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\times 49\times 36}}{2\times 49}$$

Square -300.

$$x=\frac{-\left(-300\right)±\sqrt{90000-4\times 49\times 36}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-\left(-300\right)±\sqrt{90000-196\times 36}}{2\times 49}$$

Multiply -196 times 36.

$$x=\frac{-\left(-300\right)±\sqrt{90000-7056}}{2\times 49}$$

Add 90000 to -7056.

$$x=\frac{-\left(-300\right)±\sqrt{82944}}{2\times 49}$$

Take the square root of 82944.

$$x=\frac{-\left(-300\right)±288}{2\times 49}$$

The opposite of -300 is 300.

$$x=\frac{300±288}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{300±288}{98}$$

Now solve the equation x=\frac{300±288}{98} when ± is plus. Add 300 to 288.

$$x=\frac{588}{98}$$

Divide 588 by 98.

$$x=6$$

Now solve the equation x=\frac{300±288}{98} when ± is minus. Subtract 288 from 300.

$$x=\frac{12}{98}$$

Reduce the fraction \frac{12}{98} to lowest terms by extracting and canceling out 2.

$$x=\frac{6}{49}$$

The equation is now solved.

$$x=6 x=\frac{6}{49}$$

Substitute 6 for x in the equation \sqrt{15-x}-\sqrt{6x}=-3.

$$\sqrt{15-6}-\sqrt{6\times 6}=-3$$

Simplify. The value x=6 satisfies the equation.

$$-3=-3$$

Substitute \frac{6}{49} for x in the equation \sqrt{15-x}-\sqrt{6x}=-3.

$$\sqrt{15-\frac{6}{49}}-\sqrt{6\times \frac{6}{49}}=-3$$

Simplify. The value x=\frac{6}{49} does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-3$$

Substitute 6 for x in the equation \sqrt{15-x}-\sqrt{6x}=-3.

$$\sqrt{15-6}-\sqrt{6\times 6}=-3$$

Simplify. The value x=6 satisfies the equation.

$$-3=-3$$

Equation \sqrt{15-x}=\sqrt{6x}-3 has a unique solution.

$$x=6$$