Solve 4sqrt{x-3}-sqrt{6x-17}=3

$$4 \sqrt { x - 3 } - \sqrt { 6 x - 17 } = 3$$

Solve for x

$$x=7$$

Solution Steps

Subtract -\sqrt{6x-17} from both sides of the equation.

$$4\sqrt{x-3}=3+\sqrt{6x-17}$$

Square both sides of the equation.

$$\left(4\sqrt{x-3}\right)^{2}=\left(3+\sqrt{6x-17}\right)^{2}$$

Expand \left(4\sqrt{x-3}\right)^{2}.

$$4^{2}\left(\sqrt{x-3}\right)^{2}=\left(3+\sqrt{6x-17}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x-3}\right)^{2}=\left(3+\sqrt{6x-17}\right)^{2}$$

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

$$16\left(x-3\right)=\left(3+\sqrt{6x-17}\right)^{2}$$

Use the distributive property to multiply 16 by x-3.

$$16x-48=\left(3+\sqrt{6x-17}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{6x-17}\right)^{2}.

$$16x-48=9+6\sqrt{6x-17}+\left(\sqrt{6x-17}\right)^{2}$$

Calculate \sqrt{6x-17} to the power of 2 and get 6x-17.

$$16x-48=9+6\sqrt{6x-17}+6x-17$$

Subtract 17 from 9 to get -8.

$$16x-48=-8+6\sqrt{6x-17}+6x$$

Subtract -8+6x from both sides of the equation.

$$16x-48-\left(-8+6x\right)=6\sqrt{6x-17}$$

To find the opposite of -8+6x, find the opposite of each term.

$$16x-48+8-6x=6\sqrt{6x-17}$$

Add -48 and 8 to get -40.

$$16x-40-6x=6\sqrt{6x-17}$$

Combine 16x and -6x to get 10x.

$$10x-40=6\sqrt{6x-17}$$

Square both sides of the equation.

$$\left(10x-40\right)^{2}=\left(6\sqrt{6x-17}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10x-40\right)^{2}.

$$100x^{2}-800x+1600=\left(6\sqrt{6x-17}\right)^{2}$$

Expand \left(6\sqrt{6x-17}\right)^{2}.

$$100x^{2}-800x+1600=6^{2}\left(\sqrt{6x-17}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$100x^{2}-800x+1600=36\left(\sqrt{6x-17}\right)^{2}$$

Calculate \sqrt{6x-17} to the power of 2 and get 6x-17.

$$100x^{2}-800x+1600=36\left(6x-17\right)$$

Use the distributive property to multiply 36 by 6x-17.

$$100x^{2}-800x+1600=216x-612$$

Subtract 216x from both sides.

$$100x^{2}-800x+1600-216x=-612$$

Combine -800x and -216x to get -1016x.

$$100x^{2}-1016x+1600=-612$$

Add 612 to both sides.

$$100x^{2}-1016x+1600+612=0$$

Add 1600 and 612 to get 2212.

$$100x^{2}-1016x+2212=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, -1016 for b, and 2212 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1016\right)±\sqrt{\left(-1016\right)^{2}-4\times 100\times 2212}}{2\times 100}$$

Square -1016.

$$x=\frac{-\left(-1016\right)±\sqrt{1032256-4\times 100\times 2212}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-\left(-1016\right)±\sqrt{1032256-400\times 2212}}{2\times 100}$$

Multiply -400 times 2212.

$$x=\frac{-\left(-1016\right)±\sqrt{1032256-884800}}{2\times 100}$$

Add 1032256 to -884800.

$$x=\frac{-\left(-1016\right)±\sqrt{147456}}{2\times 100}$$

Take the square root of 147456.

$$x=\frac{-\left(-1016\right)±384}{2\times 100}$$

The opposite of -1016 is 1016.

$$x=\frac{1016±384}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{1016±384}{200}$$

Now solve the equation x=\frac{1016±384}{200} when ± is plus. Add 1016 to 384.

$$x=\frac{1400}{200}$$

Divide 1400 by 200.

$$x=7$$

Now solve the equation x=\frac{1016±384}{200} when ± is minus. Subtract 384 from 1016.

$$x=\frac{632}{200}$$

Reduce the fraction \frac{632}{200} to lowest terms by extracting and canceling out 8.

$$x=\frac{79}{25}$$

The equation is now solved.

$$x=7 x=\frac{79}{25}$$

Substitute 7 for x in the equation 4\sqrt{x-3}-\sqrt{6x-17}=3.

$$4\sqrt{7-3}-\sqrt{6\times 7-17}=3$$

Simplify. The value x=7 satisfies the equation.

$$3=3$$

Substitute \frac{79}{25} for x in the equation 4\sqrt{x-3}-\sqrt{6x-17}=3.

$$4\sqrt{\frac{79}{25}-3}-\sqrt{6\times \frac{79}{25}-17}=3$$

Simplify. The value x=\frac{79}{25} does not satisfy the equation.

$$\frac{1}{5}=3$$

Substitute 7 for x in the equation 4\sqrt{x-3}-\sqrt{6x-17}=3.

$$4\sqrt{7-3}-\sqrt{6\times 7-17}=3$$

Simplify. The value x=7 satisfies the equation.

$$3=3$$

Equation 4\sqrt{x-3}=\sqrt{6x-17}+3 has a unique solution.

$$x=7$$