Solve sqrt{2x-3}+4=14-sqrt{8x-12}

$$\sqrt { 2 x - 3 } + 4 = 14 - \sqrt { 8 x - 12 }$$

Solve for x

$$x = \frac{127}{18} = 7\frac{1}{18} \approx 7.055555556$$

Solution Steps

Subtract 4 from both sides of the equation.

$$\sqrt{2x-3}=14-\sqrt{8x-12}-4$$

Subtract 4 from 14 to get 10.

$$\sqrt{2x-3}=10-\sqrt{8x-12}$$

Square both sides of the equation.

$$\left(\sqrt{2x-3}\right)^{2}=\left(10-\sqrt{8x-12}\right)^{2}$$

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

$$2x-3=\left(10-\sqrt{8x-12}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-\sqrt{8x-12}\right)^{2}.

$$2x-3=100-20\sqrt{8x-12}+\left(\sqrt{8x-12}\right)^{2}$$

Calculate \sqrt{8x-12} to the power of 2 and get 8x-12.

$$2x-3=100-20\sqrt{8x-12}+8x-12$$

Subtract 12 from 100 to get 88.

$$2x-3=88-20\sqrt{8x-12}+8x$$

Subtract 88+8x from both sides of the equation.

$$2x-3-\left(88+8x\right)=-20\sqrt{8x-12}$$

To find the opposite of 88+8x, find the opposite of each term.

$$2x-3-88-8x=-20\sqrt{8x-12}$$

Subtract 88 from -3 to get -91.

$$2x-91-8x=-20\sqrt{8x-12}$$

Combine 2x and -8x to get -6x.

$$-6x-91=-20\sqrt{8x-12}$$

Square both sides of the equation.

$$\left(-6x-91\right)^{2}=\left(-20\sqrt{8x-12}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-6x-91\right)^{2}.

$$36x^{2}+1092x+8281=\left(-20\sqrt{8x-12}\right)^{2}$$

Expand \left(-20\sqrt{8x-12}\right)^{2}.

$$36x^{2}+1092x+8281=\left(-20\right)^{2}\left(\sqrt{8x-12}\right)^{2}$$

Calculate -20 to the power of 2 and get 400.

$$36x^{2}+1092x+8281=400\left(\sqrt{8x-12}\right)^{2}$$

Calculate \sqrt{8x-12} to the power of 2 and get 8x-12.

$$36x^{2}+1092x+8281=400\left(8x-12\right)$$

Use the distributive property to multiply 400 by 8x-12.

$$36x^{2}+1092x+8281=3200x-4800$$

Subtract 3200x from both sides.

$$36x^{2}+1092x+8281-3200x=-4800$$

Combine 1092x and -3200x to get -2108x.

$$36x^{2}-2108x+8281=-4800$$

Add 4800 to both sides.

$$36x^{2}-2108x+8281+4800=0$$

Add 8281 and 4800 to get 13081.

$$36x^{2}-2108x+13081=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -2108 for b, and 13081 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2108\right)±\sqrt{\left(-2108\right)^{2}-4\times 36\times 13081}}{2\times 36}$$

Square -2108.

$$x=\frac{-\left(-2108\right)±\sqrt{4443664-4\times 36\times 13081}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-2108\right)±\sqrt{4443664-144\times 13081}}{2\times 36}$$

Multiply -144 times 13081.

$$x=\frac{-\left(-2108\right)±\sqrt{4443664-1883664}}{2\times 36}$$

Add 4443664 to -1883664.

$$x=\frac{-\left(-2108\right)±\sqrt{2560000}}{2\times 36}$$

Take the square root of 2560000.

$$x=\frac{-\left(-2108\right)±1600}{2\times 36}$$

The opposite of -2108 is 2108.

$$x=\frac{2108±1600}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{2108±1600}{72}$$

Now solve the equation x=\frac{2108±1600}{72} when ± is plus. Add 2108 to 1600.

$$x=\frac{3708}{72}$$

Reduce the fraction \frac{3708}{72} to lowest terms by extracting and canceling out 36.

$$x=\frac{103}{2}$$

Now solve the equation x=\frac{2108±1600}{72} when ± is minus. Subtract 1600 from 2108.

$$x=\frac{508}{72}$$

Reduce the fraction \frac{508}{72} to lowest terms by extracting and canceling out 4.

$$x=\frac{127}{18}$$

The equation is now solved.

$$x=\frac{103}{2} x=\frac{127}{18}$$

Substitute \frac{103}{2} for x in the equation \sqrt{2x-3}+4=14-\sqrt{8x-12}.

$$\sqrt{2\times \frac{103}{2}-3}+4=14-\sqrt{8\times \frac{103}{2}-12}$$

Simplify. The value x=\frac{103}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$14=-6$$

Substitute \frac{127}{18} for x in the equation \sqrt{2x-3}+4=14-\sqrt{8x-12}.

$$\sqrt{2\times \frac{127}{18}-3}+4=14-\sqrt{8\times \frac{127}{18}-12}$$

Simplify. The value x=\frac{127}{18} satisfies the equation.

$$\frac{22}{3}=\frac{22}{3}$$

Equation \sqrt{2x-3}=-\sqrt{8x-12}+10 has a unique solution.

$$x=\frac{127}{18}$$