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

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

Solve for x

$$x=6$$

Solution Steps

Subtract 3\sqrt{2x-3} from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{7-x}\right)^{2}=\left(11-3\sqrt{2x-3}\right)^{2}$$

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

$$4\left(7-x\right)=\left(11-3\sqrt{2x-3}\right)^{2}$$

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

$$28-4x=\left(11-3\sqrt{2x-3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-3\sqrt{2x-3}\right)^{2}.

$$28-4x=121-66\sqrt{2x-3}+9\left(\sqrt{2x-3}\right)^{2}$$

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

$$28-4x=121-66\sqrt{2x-3}+9\left(2x-3\right)$$

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

$$28-4x=121-66\sqrt{2x-3}+18x-27$$

Subtract 27 from 121 to get 94.

$$28-4x=94-66\sqrt{2x-3}+18x$$

Subtract 94+18x from both sides of the equation.

$$28-4x-\left(94+18x\right)=-66\sqrt{2x-3}$$

To find the opposite of 94+18x, find the opposite of each term.

$$28-4x-94-18x=-66\sqrt{2x-3}$$

Subtract 94 from 28 to get -66.

$$-66-4x-18x=-66\sqrt{2x-3}$$

Combine -4x and -18x to get -22x.

$$-66-22x=-66\sqrt{2x-3}$$

Square both sides of the equation.

$$\left(-66-22x\right)^{2}=\left(-66\sqrt{2x-3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-66-22x\right)^{2}.

$$4356+2904x+484x^{2}=\left(-66\sqrt{2x-3}\right)^{2}$$

Expand \left(-66\sqrt{2x-3}\right)^{2}.

$$4356+2904x+484x^{2}=\left(-66\right)^{2}\left(\sqrt{2x-3}\right)^{2}$$

Calculate -66 to the power of 2 and get 4356.

$$4356+2904x+484x^{2}=4356\left(\sqrt{2x-3}\right)^{2}$$

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

$$4356+2904x+484x^{2}=4356\left(2x-3\right)$$

Use the distributive property to multiply 4356 by 2x-3.

$$4356+2904x+484x^{2}=8712x-13068$$

Subtract 8712x from both sides.

$$4356+2904x+484x^{2}-8712x=-13068$$

Combine 2904x and -8712x to get -5808x.

$$4356-5808x+484x^{2}=-13068$$

Add 13068 to both sides.

$$4356-5808x+484x^{2}+13068=0$$

Add 4356 and 13068 to get 17424.

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

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.

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

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

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

Simplify. The value x=6 satisfies the equation.

$$11=11$$

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

$$x=6$$