Solve y^2-32.1y-15000=0

$$y ^ { 2 } - 32.1 y - 15000 = 0$$

Solve for y

$$y = \frac{\sqrt{6103041} + 321}{20} \approx 139.57166814$$

$$y=\frac{321-\sqrt{6103041}}{20}\approx -107.47166814$$

Steps Using the Quadratic Formula

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.

$$y^{2}-32.1y-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -32.1 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-32.1\right)±\sqrt{\left(-32.1\right)^{2}-4\left(-15000\right)}}{2}$$

Square -32.1 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-32.1\right)±\sqrt{1030.41-4\left(-15000\right)}}{2}$$

Multiply -4 times -15000.

$$y=\frac{-\left(-32.1\right)±\sqrt{1030.41+60000}}{2}$$

Add 1030.41 to 60000.

$$y=\frac{-\left(-32.1\right)±\sqrt{61030.41}}{2}$$

Take the square root of 61030.41.

$$y=\frac{-\left(-32.1\right)±\frac{\sqrt{6103041}}{10}}{2}$$

The opposite of -32.1 is 32.1.

$$y=\frac{32.1±\frac{\sqrt{6103041}}{10}}{2}$$

Now solve the equation y=\frac{32.1±\frac{\sqrt{6103041}}{10}}{2} when ± is plus. Add 32.1 to \frac{\sqrt{6103041}}{10}.

$$y=\frac{\sqrt{6103041}+321}{2\times 10}$$

Divide \frac{321+\sqrt{6103041}}{10} by 2.

$$y=\frac{\sqrt{6103041}+321}{20}$$

Now solve the equation y=\frac{32.1±\frac{\sqrt{6103041}}{10}}{2} when ± is minus. Subtract \frac{\sqrt{6103041}}{10} from 32.1.

$$y=\frac{321-\sqrt{6103041}}{2\times 10}$$

Divide \frac{321-\sqrt{6103041}}{10} by 2.

$$y=\frac{321-\sqrt{6103041}}{20}$$

The equation is now solved.

$$y=\frac{\sqrt{6103041}+321}{20} y=\frac{321-\sqrt{6103041}}{20}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$y^{2}-32.1y-15000=0$$

Add 15000 to both sides of the equation.

$$y^{2}-32.1y-15000-\left(-15000\right)=-\left(-15000\right)$$

Subtracting -15000 from itself leaves 0.

$$y^{2}-32.1y=-\left(-15000\right)$$

Subtract -15000 from 0.

$$y^{2}-32.1y=15000$$

Divide -32.1, the coefficient of the x term, by 2 to get -16.05. Then add the square of -16.05 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-32.1y+\left(-16.05\right)^{2}=15000+\left(-16.05\right)^{2}$$

Square -16.05 by squaring both the numerator and the denominator of the fraction.

$$y^{2}-32.1y+257.6025=15000+257.6025$$

Add 15000 to 257.6025.

$$y^{2}-32.1y+257.6025=15257.6025$$

Factor y^{2}-32.1y+257.6025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-16.05\right)^{2}=15257.6025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-16.05\right)^{2}}=\sqrt{15257.6025}$$

Simplify.

$$y-16.05=\frac{\sqrt{6103041}}{20} y-16.05=-\frac{\sqrt{6103041}}{20}$$

Add 16.05 to both sides of the equation.

$$y=\frac{\sqrt{6103041}+321}{20} y=\frac{321-\sqrt{6103041}}{20}$$