Solve (6-2sqrt{x})^2+6^2=8x

$$( 6 - 2 \sqrt { x } ) ^ { 2 } + 6 ^ { 2 } = 8 x$$

Solve for x

$$x=36-18\sqrt{3}\approx 4.823085464$$

Solution Steps

Calculate 6 to the power of 2 and get 36.

$$\left(6-2\sqrt{x}\right)^{2}+36=8x$$

Subtract 8x from both sides.

$$\left(6-2\sqrt{x}\right)^{2}+36-8x=0$$

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

$$36-24\sqrt{x}+4\left(\sqrt{x}\right)^{2}+36-8x=0$$

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

$$36-24\sqrt{x}+4x+36-8x=0$$

Add 36 and 36 to get 72.

$$72-24\sqrt{x}+4x-8x=0$$

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

$$72-24\sqrt{x}-4x=0$$

Subtract 72 from both sides. Anything subtracted from zero gives its negation.

$$-24\sqrt{x}-4x=-72$$

Subtract -4x from both sides of the equation.

$$-24\sqrt{x}=-72+4x$$

Square both sides of the equation.

$$\left(-24\sqrt{x}\right)^{2}=\left(4x-72\right)^{2}$$

Expand \left(-24\sqrt{x}\right)^{2}.

$$\left(-24\right)^{2}\left(\sqrt{x}\right)^{2}=\left(4x-72\right)^{2}$$

Calculate -24 to the power of 2 and get 576.

$$576\left(\sqrt{x}\right)^{2}=\left(4x-72\right)^{2}$$

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

$$576x=\left(4x-72\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-72\right)^{2}.

$$576x=16x^{2}-576x+5184$$

Subtract 16x^{2} from both sides.

$$576x-16x^{2}=-576x+5184$$

Add 576x to both sides.

$$576x-16x^{2}+576x=5184$$

Combine 576x and 576x to get 1152x.

$$1152x-16x^{2}=5184$$

Subtract 5184 from both sides.

$$1152x-16x^{2}-5184=0$$

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.

$$-16x^{2}+1152x-5184=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 1152 for b, and -5184 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1152±\sqrt{1152^{2}-4\left(-16\right)\left(-5184\right)}}{2\left(-16\right)}$$

Square 1152.

$$x=\frac{-1152±\sqrt{1327104-4\left(-16\right)\left(-5184\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-1152±\sqrt{1327104+64\left(-5184\right)}}{2\left(-16\right)}$$

Multiply 64 times -5184.

$$x=\frac{-1152±\sqrt{1327104-331776}}{2\left(-16\right)}$$

Add 1327104 to -331776.

$$x=\frac{-1152±\sqrt{995328}}{2\left(-16\right)}$$

Take the square root of 995328.

$$x=\frac{-1152±576\sqrt{3}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-1152±576\sqrt{3}}{-32}$$

Now solve the equation x=\frac{-1152±576\sqrt{3}}{-32} when ± is plus. Add -1152 to 576\sqrt{3}.

$$x=\frac{576\sqrt{3}-1152}{-32}$$

Divide -1152+576\sqrt{3} by -32.

$$x=36-18\sqrt{3}$$

Now solve the equation x=\frac{-1152±576\sqrt{3}}{-32} when ± is minus. Subtract 576\sqrt{3} from -1152.

$$x=\frac{-576\sqrt{3}-1152}{-32}$$

Divide -1152-576\sqrt{3} by -32.

$$x=18\sqrt{3}+36$$

The equation is now solved.

$$x=36-18\sqrt{3} x=18\sqrt{3}+36$$

Substitute 36-18\sqrt{3} for x in the equation \left(6-2\sqrt{x}\right)^{2}+6^{2}=8x.

$$\left(6-2\sqrt{36-18\sqrt{3}}\right)^{2}+6^{2}=8\left(36-18\sqrt{3}\right)$$

Simplify. The value x=36-18\sqrt{3} satisfies the equation.

$$288-144\times 3^{\frac{1}{2}}=288-144\times 3^{\frac{1}{2}}$$

Substitute 18\sqrt{3}+36 for x in the equation \left(6-2\sqrt{x}\right)^{2}+6^{2}=8x.

$$\left(6-2\sqrt{18\sqrt{3}+36}\right)^{2}+6^{2}=8\left(18\sqrt{3}+36\right)$$

Simplify. The value x=18\sqrt{3}+36 does not satisfy the equation.

$$144=144\times 3^{\frac{1}{2}}+288$$

Equation -24\sqrt{x}=4x-72 has a unique solution.

$$x=36-18\sqrt{3}$$