Solve 10*2+1/2*{left(2sqrt{7}right)}^2=10*8/3+1/2*x^2

$$10 \times 2+ \frac{ 1 }{ 2 } \times { \left(2 \sqrt{ 7 } \right) }^{ 2 } =10 \times \frac{ 8 }{ 3 } + \frac{ 1 }{ 2 } \times { x }^{ 2 }$$

Solve for x

$$x = \frac{2 \sqrt{33}}{3} \approx 3.829708431$$

$$x = -\frac{2 \sqrt{33}}{3} \approx -3.829708431$$

Steps by Finding Square Root

Multiply 10 and 2 to get 20.

$$20+\frac{1}{2}\times \left(2\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

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

$$20+\frac{1}{2}\times 2^{2}\left(\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$20+\frac{1}{2}\times 4\left(\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

The square of \sqrt{7} is 7.

$$20+\frac{1}{2}\times 4\times 7=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply 4 and 7 to get 28.

$$20+\frac{1}{2}\times 28=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply \frac{1}{2} and 28 to get 14.

$$20+14=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Add 20 and 14 to get 34.

$$34=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply 10 and \frac{8}{3} to get \frac{80}{3}.

$$34=\frac{80}{3}+\frac{1}{2}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{80}{3}+\frac{1}{2}x^{2}=34$$

Subtract \frac{80}{3} from both sides.

$$\frac{1}{2}x^{2}=34-\frac{80}{3}$$

Subtract \frac{80}{3} from 34 to get \frac{22}{3}.

$$\frac{1}{2}x^{2}=\frac{22}{3}$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{22}{3}\times 2$$

Multiply \frac{22}{3} and 2 to get \frac{44}{3}.

$$x^{2}=\frac{44}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{2\sqrt{33}}{3} x=-\frac{2\sqrt{33}}{3}$$

Steps Using the Quadratic Formula

Multiply 10 and 2 to get 20.

$$20+\frac{1}{2}\times \left(2\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

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

$$20+\frac{1}{2}\times 2^{2}\left(\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$20+\frac{1}{2}\times 4\left(\sqrt{7}\right)^{2}=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

The square of \sqrt{7} is 7.

$$20+\frac{1}{2}\times 4\times 7=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply 4 and 7 to get 28.

$$20+\frac{1}{2}\times 28=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply \frac{1}{2} and 28 to get 14.

$$20+14=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Add 20 and 14 to get 34.

$$34=10\times \frac{8}{3}+\frac{1}{2}x^{2}$$

Multiply 10 and \frac{8}{3} to get \frac{80}{3}.

$$34=\frac{80}{3}+\frac{1}{2}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{80}{3}+\frac{1}{2}x^{2}=34$$

Subtract 34 from both sides.

$$\frac{80}{3}+\frac{1}{2}x^{2}-34=0$$

Subtract 34 from \frac{80}{3} to get -\frac{22}{3}.

$$-\frac{22}{3}+\frac{1}{2}x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\frac{1}{2}x^{2}-\frac{22}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 0 for b, and -\frac{22}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\left(-\frac{22}{3}\right)}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\left(-\frac{22}{3}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\left(-\frac{22}{3}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{22}{3}.

$$x=\frac{0±\sqrt{\frac{44}{3}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{44}{3}.

$$x=\frac{0±\frac{2\sqrt{33}}{3}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{0±\frac{2\sqrt{33}}{3}}{1}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{33}}{3}}{1} when ± is plus.

$$x=\frac{2\sqrt{33}}{3}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{33}}{3}}{1} when ± is minus.

$$x=-\frac{2\sqrt{33}}{3}$$

The equation is now solved.

$$x=\frac{2\sqrt{33}}{3} x=-\frac{2\sqrt{33}}{3}$$