Solve 3+sqrt{x-3}=4-sqrt{4x-12}

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

Solve for x

$$x = \frac{28}{9} = 3\frac{1}{9} \approx 3.111111111$$

Solution Steps

Subtract 3 from both sides of the equation.

$$\sqrt{x-3}=4-\sqrt{4x-12}-3$$

Subtract 3 from 4 to get 1.

$$\sqrt{x-3}=1-\sqrt{4x-12}$$

Square both sides of the equation.

$$\left(\sqrt{x-3}\right)^{2}=\left(1-\sqrt{4x-12}\right)^{2}$$

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

$$x-3=\left(1-\sqrt{4x-12}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{4x-12}\right)^{2}.

$$x-3=1-2\sqrt{4x-12}+\left(\sqrt{4x-12}\right)^{2}$$

Calculate \sqrt{4x-12} to the power of 2 and get 4x-12.

$$x-3=1-2\sqrt{4x-12}+4x-12$$

Subtract 12 from 1 to get -11.

$$x-3=-11-2\sqrt{4x-12}+4x$$

Subtract -11+4x from both sides of the equation.

$$x-3-\left(-11+4x\right)=-2\sqrt{4x-12}$$

To find the opposite of -11+4x, find the opposite of each term.

$$x-3+11-4x=-2\sqrt{4x-12}$$

Add -3 and 11 to get 8.

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

Combine x and -4x to get -3x.

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

Square both sides of the equation.

$$\left(-3x+8\right)^{2}=\left(-2\sqrt{4x-12}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3x+8\right)^{2}.

$$9x^{2}-48x+64=\left(-2\sqrt{4x-12}\right)^{2}$$

Expand \left(-2\sqrt{4x-12}\right)^{2}.

$$9x^{2}-48x+64=\left(-2\right)^{2}\left(\sqrt{4x-12}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$9x^{2}-48x+64=4\left(\sqrt{4x-12}\right)^{2}$$

Calculate \sqrt{4x-12} to the power of 2 and get 4x-12.

$$9x^{2}-48x+64=4\left(4x-12\right)$$

Use the distributive property to multiply 4 by 4x-12.

$$9x^{2}-48x+64=16x-48$$

Subtract 16x from both sides.

$$9x^{2}-48x+64-16x=-48$$

Combine -48x and -16x to get -64x.

$$9x^{2}-64x+64=-48$$

Add 48 to both sides.

$$9x^{2}-64x+64+48=0$$

Add 64 and 48 to get 112.

$$9x^{2}-64x+112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -64 for b, and 112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 9\times 112}}{2\times 9}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 9\times 112}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-64\right)±\sqrt{4096-36\times 112}}{2\times 9}$$

Multiply -36 times 112.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4032}}{2\times 9}$$

Add 4096 to -4032.

$$x=\frac{-\left(-64\right)±\sqrt{64}}{2\times 9}$$

Take the square root of 64.

$$x=\frac{-\left(-64\right)±8}{2\times 9}$$

The opposite of -64 is 64.

$$x=\frac{64±8}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{64±8}{18}$$

Now solve the equation x=\frac{64±8}{18} when ± is plus. Add 64 to 8.

$$x=\frac{72}{18}$$

Divide 72 by 18.

$$x=4$$

Now solve the equation x=\frac{64±8}{18} when ± is minus. Subtract 8 from 64.

$$x=\frac{56}{18}$$

Reduce the fraction \frac{56}{18} to lowest terms by extracting and canceling out 2.

$$x=\frac{28}{9}$$

The equation is now solved.

$$x=4 x=\frac{28}{9}$$

Substitute 4 for x in the equation 3+\sqrt{x-3}=4-\sqrt{4x-12}.

$$3+\sqrt{4-3}=4-\sqrt{4\times 4-12}$$

Simplify. The value x=4 does not satisfy the equation.

$$4=2$$

Substitute \frac{28}{9} for x in the equation 3+\sqrt{x-3}=4-\sqrt{4x-12}.

$$3+\sqrt{\frac{28}{9}-3}=4-\sqrt{4\times \frac{28}{9}-12}$$

Simplify. The value x=\frac{28}{9} satisfies the equation.

$$\frac{10}{3}=\frac{10}{3}$$

Equation \sqrt{x-3}=-\sqrt{4x-12}+1 has a unique solution.

$$x=\frac{28}{9}$$