Solve 3sqrt{2x-3}+2sqrt{7-x}=11

$$3 \sqrt{ 2x-3 } +2 \sqrt{ 7-x } =11$$

Solve for x

$$x=6$$

Solution Steps

Subtract 2\sqrt{7-x} from both sides of the equation.

$$3\sqrt{2x-3}=11-2\sqrt{7-x}$$

Square both sides of the equation.

$$\left(3\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

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

$$3^{2}\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

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

$$9\left(2x-3\right)=\left(11-2\sqrt{7-x}\right)^{2}$$

Use the distributive property to multiply 9 by 2x-3.

$$18x-27=\left(11-2\sqrt{7-x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-2\sqrt{7-x}\right)^{2}.

$$18x-27=121-44\sqrt{7-x}+4\left(\sqrt{7-x}\right)^{2}$$

Calculate \sqrt{7-x} to the power of 2 and get 7-x.

$$18x-27=121-44\sqrt{7-x}+4\left(7-x\right)$$

Use the distributive property to multiply 4 by 7-x.

$$18x-27=121-44\sqrt{7-x}+28-4x$$

Add 121 and 28 to get 149.

$$18x-27=149-44\sqrt{7-x}-4x$$

Subtract 149-4x from both sides of the equation.

$$18x-27-\left(149-4x\right)=-44\sqrt{7-x}$$

To find the opposite of 149-4x, find the opposite of each term.

$$18x-27-149+4x=-44\sqrt{7-x}$$

Subtract 149 from -27 to get -176.

$$18x-176+4x=-44\sqrt{7-x}$$

Combine 18x and 4x to get 22x.

$$22x-176=-44\sqrt{7-x}$$

Square both sides of the equation.

$$\left(22x-176\right)^{2}=\left(-44\sqrt{7-x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(22x-176\right)^{2}.

$$484x^{2}-7744x+30976=\left(-44\sqrt{7-x}\right)^{2}$$

Expand \left(-44\sqrt{7-x}\right)^{2}.

$$484x^{2}-7744x+30976=\left(-44\right)^{2}\left(\sqrt{7-x}\right)^{2}$$

Calculate -44 to the power of 2 and get 1936.

$$484x^{2}-7744x+30976=1936\left(\sqrt{7-x}\right)^{2}$$

Calculate \sqrt{7-x} to the power of 2 and get 7-x.

$$484x^{2}-7744x+30976=1936\left(7-x\right)$$

Use the distributive property to multiply 1936 by 7-x.

$$484x^{2}-7744x+30976=13552-1936x$$

Subtract 13552 from both sides.

$$484x^{2}-7744x+30976-13552=-1936x$$

Subtract 13552 from 30976 to get 17424.

$$484x^{2}-7744x+17424=-1936x$$

Add 1936x to both sides.

$$484x^{2}-7744x+17424+1936x=0$$

Combine -7744x and 1936x to get -5808x.

$$484x^{2}-5808x+17424=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 484 for a, -5808 for b, and 17424 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5808\right)±\sqrt{\left(-5808\right)^{2}-4\times 484\times 17424}}{2\times 484}$$

Square -5808.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-4\times 484\times 17424}}{2\times 484}$$

Multiply -4 times 484.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-1936\times 17424}}{2\times 484}$$

Multiply -1936 times 17424.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-33732864}}{2\times 484}$$

Add 33732864 to -33732864.

$$x=\frac{-\left(-5808\right)±\sqrt{0}}{2\times 484}$$

Take the square root of 0.

$$x=-\frac{-5808}{2\times 484}$$

The opposite of -5808 is 5808.

$$x=\frac{5808}{2\times 484}$$

Multiply 2 times 484.

$$x=\frac{5808}{968}$$

Divide 5808 by 968.

$$x=6$$

Substitute 6 for x in the equation 3\sqrt{2x-3}+2\sqrt{7-x}=11.

$$3\sqrt{2\times 6-3}+2\sqrt{7-6}=11$$

Simplify. The value x=6 satisfies the equation.

$$11=11$$

Equation 3\sqrt{2x-3}=-2\sqrt{7-x}+11 has a unique solution.

$$x=6$$