Solve x^2={left(6sqrt{3-24}right)}^2-{left(3sqrt{3}-12right)}^2

$${ x }^{ 2 } = { \left(6 \sqrt{ 3-24 } \right) }^{ 2 } - { \left(3 \sqrt{ 3 } -12 \right) }^{ 2 }$$

Solve for x (complex solution)

$$x=-3i\sqrt{103-8\sqrt{3}}\approx -0-28.324765522i$$

$$x=3i\sqrt{103-8\sqrt{3}}\approx 28.324765522i$$

Steps by Finding Square Root

Subtract 24 from 3 to get -21.

$$x^{2}=\left(6\sqrt{-21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Factor -21=21\left(-1\right). Rewrite the square root of the product \sqrt{21\left(-1\right)} as the product of square roots \sqrt{21}\sqrt{-1}. By definition, the square root of -1 is i.

$$x^{2}=\left(6\sqrt{21}i\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Multiply 6 and i to get 6i.

$$x^{2}=\left(6i\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Expand \left(6i\sqrt{21}\right)^{2}.

$$x^{2}=\left(6i\right)^{2}\left(\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Calculate 6i to the power of 2 and get -36.

$$x^{2}=-36\left(\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

The square of \sqrt{21} is 21.

$$x^{2}=-36\times 21-\left(3\sqrt{3}-12\right)^{2}$$

Multiply -36 and 21 to get -756.

$$x^{2}=-756-\left(3\sqrt{3}-12\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{3}-12\right)^{2}.

$$x^{2}=-756-\left(9\left(\sqrt{3}\right)^{2}-72\sqrt{3}+144\right)$$

The square of \sqrt{3} is 3.

$$x^{2}=-756-\left(9\times 3-72\sqrt{3}+144\right)$$

Multiply 9 and 3 to get 27.

$$x^{2}=-756-\left(27-72\sqrt{3}+144\right)$$

Add 27 and 144 to get 171.

$$x^{2}=-756-\left(171-72\sqrt{3}\right)$$

To find the opposite of 171-72\sqrt{3}, find the opposite of each term.

$$x^{2}=-756-171+72\sqrt{3}$$

Subtract 171 from -756 to get -927.

$$x^{2}=-927+72\sqrt{3}$$

The equation is now solved.

$$x=3i\sqrt{103-8\sqrt{3}} x=-3i\sqrt{103-8\sqrt{3}}$$

Steps Using the Quadratic Formula

Subtract 24 from 3 to get -21.

$$x^{2}=\left(6\sqrt{-21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Factor -21=21\left(-1\right). Rewrite the square root of the product \sqrt{21\left(-1\right)} as the product of square roots \sqrt{21}\sqrt{-1}. By definition, the square root of -1 is i.

$$x^{2}=\left(6\sqrt{21}i\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Multiply 6 and i to get 6i.

$$x^{2}=\left(6i\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Expand \left(6i\sqrt{21}\right)^{2}.

$$x^{2}=\left(6i\right)^{2}\left(\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

Calculate 6i to the power of 2 and get -36.

$$x^{2}=-36\left(\sqrt{21}\right)^{2}-\left(3\sqrt{3}-12\right)^{2}$$

The square of \sqrt{21} is 21.

$$x^{2}=-36\times 21-\left(3\sqrt{3}-12\right)^{2}$$

Multiply -36 and 21 to get -756.

$$x^{2}=-756-\left(3\sqrt{3}-12\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{3}-12\right)^{2}.

$$x^{2}=-756-\left(9\left(\sqrt{3}\right)^{2}-72\sqrt{3}+144\right)$$

The square of \sqrt{3} is 3.

$$x^{2}=-756-\left(9\times 3-72\sqrt{3}+144\right)$$

Multiply 9 and 3 to get 27.

$$x^{2}=-756-\left(27-72\sqrt{3}+144\right)$$

Add 27 and 144 to get 171.

$$x^{2}=-756-\left(171-72\sqrt{3}\right)$$

To find the opposite of 171-72\sqrt{3}, find the opposite of each term.

$$x^{2}=-756-171+72\sqrt{3}$$

Subtract 171 from -756 to get -927.

$$x^{2}=-927+72\sqrt{3}$$

Subtract -927 from both sides.

$$x^{2}-\left(-927\right)=72\sqrt{3}$$

The opposite of -927 is 927.

$$x^{2}+927=72\sqrt{3}$$

Subtract 72\sqrt{3} from both sides.

$$x^{2}+927-72\sqrt{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and 927-72\sqrt{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(927-72\sqrt{3}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(927-72\sqrt{3}\right)}}{2}$$

Multiply -4 times 927-72\sqrt{3}.

$$x=\frac{0±\sqrt{288\sqrt{3}-3708}}{2}$$

Take the square root of -3708+288\sqrt{3}.

$$x=\frac{0±6i\sqrt{103-8\sqrt{3}}}{2}$$

Now solve the equation x=\frac{0±6i\sqrt{103-8\sqrt{3}}}{2} when ± is plus.

$$x=3i\sqrt{103-8\sqrt{3}}$$

Now solve the equation x=\frac{0±6i\sqrt{103-8\sqrt{3}}}{2} when ± is minus.

$$x=-3i\sqrt{103-8\sqrt{3}}$$

The equation is now solved.

$$x=3i\sqrt{103-8\sqrt{3}} x=-3i\sqrt{103-8\sqrt{3}}$$