Solve 1/3*10*(x^2-10x+100)=125

$$\frac { 1 } { 3 } \times 10 \times ( x ^ { 2 } - 10 x + 100 ) = 125$$

Solve for x (complex solution)

$$x=\frac{5\sqrt{6}i}{2}+5\approx 5+6.123724357i$$

$$x=-\frac{5\sqrt{6}i}{2}+5\approx 5-6.123724357i$$

Steps Using the Quadratic Formula

Multiply \frac{1}{3} and 10 to get \frac{10}{3}.

$$\frac{10}{3}\left(x^{2}-10x+100\right)=125$$

Use the distributive property to multiply \frac{10}{3} by x^{2}-10x+100.

$$\frac{10}{3}x^{2}-\frac{100}{3}x+\frac{1000}{3}=125$$

Subtract 125 from both sides.

$$\frac{10}{3}x^{2}-\frac{100}{3}x+\frac{1000}{3}-125=0$$

Subtract 125 from \frac{1000}{3} to get \frac{625}{3}.

$$\frac{10}{3}x^{2}-\frac{100}{3}x+\frac{625}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{10}{3} for a, -\frac{100}{3} for b, and \frac{625}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{100}{3}\right)±\sqrt{\left(-\frac{100}{3}\right)^{2}-4\times \frac{10}{3}\times \frac{625}{3}}}{2\times \frac{10}{3}}$$

Square -\frac{100}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{100}{3}\right)±\sqrt{\frac{10000}{9}-4\times \frac{10}{3}\times \frac{625}{3}}}{2\times \frac{10}{3}}$$

Multiply -4 times \frac{10}{3}.

$$x=\frac{-\left(-\frac{100}{3}\right)±\sqrt{\frac{10000}{9}-\frac{40}{3}\times \frac{625}{3}}}{2\times \frac{10}{3}}$$

Multiply -\frac{40}{3} times \frac{625}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{100}{3}\right)±\sqrt{\frac{10000-25000}{9}}}{2\times \frac{10}{3}}$$

Add \frac{10000}{9} to -\frac{25000}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{100}{3}\right)±\sqrt{-\frac{5000}{3}}}{2\times \frac{10}{3}}$$

Take the square root of -\frac{5000}{3}.

$$x=\frac{-\left(-\frac{100}{3}\right)±\frac{50\sqrt{6}i}{3}}{2\times \frac{10}{3}}$$

The opposite of -\frac{100}{3} is \frac{100}{3}.

$$x=\frac{\frac{100}{3}±\frac{50\sqrt{6}i}{3}}{2\times \frac{10}{3}}$$

Multiply 2 times \frac{10}{3}.

$$x=\frac{\frac{100}{3}±\frac{50\sqrt{6}i}{3}}{\frac{20}{3}}$$

Now solve the equation x=\frac{\frac{100}{3}±\frac{50\sqrt{6}i}{3}}{\frac{20}{3}} when ± is plus. Add \frac{100}{3} to \frac{50i\sqrt{6}}{3}.

$$x=\frac{100+50\sqrt{6}i}{3\times \frac{20}{3}}$$

Divide \frac{100+50i\sqrt{6}}{3} by \frac{20}{3} by multiplying \frac{100+50i\sqrt{6}}{3} by the reciprocal of \frac{20}{3}.

$$x=\frac{5\sqrt{6}i}{2}+5$$

Now solve the equation x=\frac{\frac{100}{3}±\frac{50\sqrt{6}i}{3}}{\frac{20}{3}} when ± is minus. Subtract \frac{50i\sqrt{6}}{3} from \frac{100}{3}.

$$x=\frac{-50\sqrt{6}i+100}{3\times \frac{20}{3}}$$

Divide \frac{100-50i\sqrt{6}}{3} by \frac{20}{3} by multiplying \frac{100-50i\sqrt{6}}{3} by the reciprocal of \frac{20}{3}.

$$x=-\frac{5\sqrt{6}i}{2}+5$$

The equation is now solved.

$$x=\frac{5\sqrt{6}i}{2}+5 x=-\frac{5\sqrt{6}i}{2}+5$$

Steps for Completing the Square

Multiply \frac{1}{3} and 10 to get \frac{10}{3}.

$$\frac{10}{3}\left(x^{2}-10x+100\right)=125$$

Use the distributive property to multiply \frac{10}{3} by x^{2}-10x+100.

$$\frac{10}{3}x^{2}-\frac{100}{3}x+\frac{1000}{3}=125$$

Subtract \frac{1000}{3} from both sides.

$$\frac{10}{3}x^{2}-\frac{100}{3}x=125-\frac{1000}{3}$$

Subtract \frac{1000}{3} from 125 to get -\frac{625}{3}.

$$\frac{10}{3}x^{2}-\frac{100}{3}x=-\frac{625}{3}$$

Divide both sides of the equation by \frac{10}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{10}{3}x^{2}-\frac{100}{3}x}{\frac{10}{3}}=-\frac{\frac{625}{3}}{\frac{10}{3}}$$

Dividing by \frac{10}{3} undoes the multiplication by \frac{10}{3}.

$$x^{2}+\left(-\frac{\frac{100}{3}}{\frac{10}{3}}\right)x=-\frac{\frac{625}{3}}{\frac{10}{3}}$$

Divide -\frac{100}{3} by \frac{10}{3} by multiplying -\frac{100}{3} by the reciprocal of \frac{10}{3}.

$$x^{2}-10x=-\frac{\frac{625}{3}}{\frac{10}{3}}$$

Divide -\frac{625}{3} by \frac{10}{3} by multiplying -\frac{625}{3} by the reciprocal of \frac{10}{3}.

$$x^{2}-10x=-\frac{125}{2}$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-\frac{125}{2}+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-\frac{125}{2}+25$$

Add -\frac{125}{2} to 25.

$$x^{2}-10x+25=-\frac{75}{2}$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=-\frac{75}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-\frac{75}{2}}$$

Simplify.

$$x-5=\frac{5\sqrt{6}i}{2} x-5=-\frac{5\sqrt{6}i}{2}$$

Add 5 to both sides of the equation.

$$x=\frac{5\sqrt{6}i}{2}+5 x=-\frac{5\sqrt{6}i}{2}+5$$