Solve 0=(7div900)*left(x-30right)^2+15

$$0=(7 \div 900) \times { \left(x-30 \right) }^{ 2 } +15$$

Solve for x (complex solution)

$$x=-\frac{30\sqrt{105}i}{7}+30\approx 30-43.915503283i$$

$$x=\frac{30\sqrt{105}i}{7}+30\approx 30+43.915503283i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-30\right)^{2}.

$$0=\frac{7}{900}\left(x^{2}-60x+900\right)+15$$

Use the distributive property to multiply \frac{7}{900} by x^{2}-60x+900.

$$0=\frac{7}{900}x^{2}-\frac{7}{15}x+7+15$$

Add 7 and 15 to get 22.

$$0=\frac{7}{900}x^{2}-\frac{7}{15}x+22$$

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

$$\frac{7}{900}x^{2}-\frac{7}{15}x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{7}{900} for a, -\frac{7}{15} for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{7}{15}\right)±\sqrt{\left(-\frac{7}{15}\right)^{2}-4\times \frac{7}{900}\times 22}}{2\times \frac{7}{900}}$$

Square -\frac{7}{15} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{7}{15}\right)±\sqrt{\frac{49}{225}-4\times \frac{7}{900}\times 22}}{2\times \frac{7}{900}}$$

Multiply -4 times \frac{7}{900}.

$$x=\frac{-\left(-\frac{7}{15}\right)±\sqrt{\frac{49}{225}-\frac{7}{225}\times 22}}{2\times \frac{7}{900}}$$

Multiply -\frac{7}{225} times 22.

$$x=\frac{-\left(-\frac{7}{15}\right)±\sqrt{\frac{49-154}{225}}}{2\times \frac{7}{900}}$$

Add \frac{49}{225} to -\frac{154}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{7}{15}\right)±\sqrt{-\frac{7}{15}}}{2\times \frac{7}{900}}$$

Take the square root of -\frac{7}{15}.

$$x=\frac{-\left(-\frac{7}{15}\right)±\frac{\sqrt{105}i}{15}}{2\times \frac{7}{900}}$$

The opposite of -\frac{7}{15} is \frac{7}{15}.

$$x=\frac{\frac{7}{15}±\frac{\sqrt{105}i}{15}}{2\times \frac{7}{900}}$$

Multiply 2 times \frac{7}{900}.

$$x=\frac{\frac{7}{15}±\frac{\sqrt{105}i}{15}}{\frac{7}{450}}$$

Now solve the equation x=\frac{\frac{7}{15}±\frac{\sqrt{105}i}{15}}{\frac{7}{450}} when ± is plus. Add \frac{7}{15} to \frac{i\sqrt{105}}{15}.

$$x=\frac{7+\sqrt{105}i}{\frac{7}{450}\times 15}$$

Divide \frac{7+i\sqrt{105}}{15} by \frac{7}{450} by multiplying \frac{7+i\sqrt{105}}{15} by the reciprocal of \frac{7}{450}.

$$x=\frac{30\sqrt{105}i}{7}+30$$

Now solve the equation x=\frac{\frac{7}{15}±\frac{\sqrt{105}i}{15}}{\frac{7}{450}} when ± is minus. Subtract \frac{i\sqrt{105}}{15} from \frac{7}{15}.

$$x=\frac{-\sqrt{105}i+7}{\frac{7}{450}\times 15}$$

Divide \frac{7-i\sqrt{105}}{15} by \frac{7}{450} by multiplying \frac{7-i\sqrt{105}}{15} by the reciprocal of \frac{7}{450}.

$$x=-\frac{30\sqrt{105}i}{7}+30$$

The equation is now solved.

$$x=\frac{30\sqrt{105}i}{7}+30 x=-\frac{30\sqrt{105}i}{7}+30$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-30\right)^{2}.

$$0=\frac{7}{900}\left(x^{2}-60x+900\right)+15$$

Use the distributive property to multiply \frac{7}{900} by x^{2}-60x+900.

$$0=\frac{7}{900}x^{2}-\frac{7}{15}x+7+15$$

Add 7 and 15 to get 22.

$$0=\frac{7}{900}x^{2}-\frac{7}{15}x+22$$

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

$$\frac{7}{900}x^{2}-\frac{7}{15}x+22=0$$

Subtract 22 from both sides. Anything subtracted from zero gives its negation.

$$\frac{7}{900}x^{2}-\frac{7}{15}x=-22$$

Divide both sides of the equation by \frac{7}{900}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{7}{900}x^{2}-\frac{7}{15}x}{\frac{7}{900}}=-\frac{22}{\frac{7}{900}}$$

Dividing by \frac{7}{900} undoes the multiplication by \frac{7}{900}.

$$x^{2}+\left(-\frac{\frac{7}{15}}{\frac{7}{900}}\right)x=-\frac{22}{\frac{7}{900}}$$

Divide -\frac{7}{15} by \frac{7}{900} by multiplying -\frac{7}{15} by the reciprocal of \frac{7}{900}.

$$x^{2}-60x=-\frac{22}{\frac{7}{900}}$$

Divide -22 by \frac{7}{900} by multiplying -22 by the reciprocal of \frac{7}{900}.

$$x^{2}-60x=-\frac{19800}{7}$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-\frac{19800}{7}+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-\frac{19800}{7}+900$$

Add -\frac{19800}{7} to 900.

$$x^{2}-60x+900=-\frac{13500}{7}$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=-\frac{13500}{7}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{-\frac{13500}{7}}$$

Simplify.

$$x-30=\frac{30\sqrt{105}i}{7} x-30=-\frac{30\sqrt{105}i}{7}$$

Add 30 to both sides of the equation.

$$x=\frac{30\sqrt{105}i}{7}+30 x=-\frac{30\sqrt{105}i}{7}+30$$