Solve sqrt{12x+4(3-x)}=12(x+2)-4

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

Solve for x (complex solution)

$$x=\frac{-59+\sqrt{11}i}{36}\approx -1.638888889+0.092128466i$$

Solution Steps

Square both sides of the equation.

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

Use the distributive property to multiply 4 by 3-x.

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

Combine 12x and -4x to get 8x.

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

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

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

Use the distributive property to multiply 12 by x+2.

$$8x+12=\left(12x+24-4\right)^{2}$$

Subtract 4 from 24 to get 20.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12x+20\right)^{2}.

$$8x+12=144x^{2}+480x+400$$

Subtract 144x^{2} from both sides.

$$8x+12-144x^{2}=480x+400$$

Subtract 480x from both sides.

$$8x+12-144x^{2}-480x=400$$

Combine 8x and -480x to get -472x.

$$-472x+12-144x^{2}=400$$

Subtract 400 from both sides.

$$-472x+12-144x^{2}-400=0$$

Subtract 400 from 12 to get -388.

$$-472x-388-144x^{2}=0$$

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.

$$-144x^{2}-472x-388=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -144 for a, -472 for b, and -388 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-472\right)±\sqrt{\left(-472\right)^{2}-4\left(-144\right)\left(-388\right)}}{2\left(-144\right)}$$

Square -472.

$$x=\frac{-\left(-472\right)±\sqrt{222784-4\left(-144\right)\left(-388\right)}}{2\left(-144\right)}$$

Multiply -4 times -144.

$$x=\frac{-\left(-472\right)±\sqrt{222784+576\left(-388\right)}}{2\left(-144\right)}$$

Multiply 576 times -388.

$$x=\frac{-\left(-472\right)±\sqrt{222784-223488}}{2\left(-144\right)}$$

Add 222784 to -223488.

$$x=\frac{-\left(-472\right)±\sqrt{-704}}{2\left(-144\right)}$$

Take the square root of -704.

$$x=\frac{-\left(-472\right)±8\sqrt{11}i}{2\left(-144\right)}$$

The opposite of -472 is 472.

$$x=\frac{472±8\sqrt{11}i}{2\left(-144\right)}$$

Multiply 2 times -144.

$$x=\frac{472±8\sqrt{11}i}{-288}$$

Now solve the equation x=\frac{472±8\sqrt{11}i}{-288} when ± is plus. Add 472 to 8i\sqrt{11}.

$$x=\frac{472+8\sqrt{11}i}{-288}$$

Divide 472+8i\sqrt{11} by -288.

$$x=\frac{-\sqrt{11}i-59}{36}$$

Now solve the equation x=\frac{472±8\sqrt{11}i}{-288} when ± is minus. Subtract 8i\sqrt{11} from 472.

$$x=\frac{-8\sqrt{11}i+472}{-288}$$

Divide 472-8i\sqrt{11} by -288.

$$x=\frac{-59+\sqrt{11}i}{36}$$

The equation is now solved.

$$x=\frac{-\sqrt{11}i-59}{36} x=\frac{-59+\sqrt{11}i}{36}$$

Substitute \frac{-\sqrt{11}i-59}{36} for x in the equation \sqrt{12x+4\left(3-x\right)}=12\left(x+2\right)-4.

$$\sqrt{12\times \frac{-\sqrt{11}i-59}{36}+4\left(3-\frac{-\sqrt{11}i-59}{36}\right)}=12\left(\frac{-\sqrt{11}i-59}{36}+2\right)-4$$

Simplify. The value x=\frac{-\sqrt{11}i-59}{36} does not satisfy the equation.

$$-\frac{1}{3}+\frac{1}{3}i\times 11^{\frac{1}{2}}=-\frac{1}{3}i\times 11^{\frac{1}{2}}+\frac{1}{3}$$

Substitute \frac{-59+\sqrt{11}i}{36} for x in the equation \sqrt{12x+4\left(3-x\right)}=12\left(x+2\right)-4.

$$\sqrt{12\times \frac{-59+\sqrt{11}i}{36}+4\left(3-\frac{-59+\sqrt{11}i}{36}\right)}=12\left(\frac{-59+\sqrt{11}i}{36}+2\right)-4$$

Simplify. The value x=\frac{-59+\sqrt{11}i}{36} satisfies the equation.

$$\frac{1}{3}+\frac{1}{3}i\times 11^{\frac{1}{2}}=\frac{1}{3}+\frac{1}{3}i\times 11^{\frac{1}{2}}$$

Equation \sqrt{4\left(3-x\right)+12x}=12\left(x+2\right)-4 has a unique solution.

$$x=\frac{-59+\sqrt{11}i}{36}$$