Solve 30t+2t^2=300

$$30 t + 2 t ^ { 2 } = 300$$

Solve for t

$$t = \frac{5 \sqrt{33} - 15}{2} \approx 6.861406616$$

$$t=\frac{-5\sqrt{33}-15}{2}\approx -21.861406616$$

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.

$$2t^{2}+30t=300$$

Subtract 300 from both sides of the equation.

$$2t^{2}+30t-300=300-300$$

Subtracting 300 from itself leaves 0.

$$2t^{2}+30t-300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 30 for b, and -300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-30±\sqrt{30^{2}-4\times 2\left(-300\right)}}{2\times 2}$$

Square 30.

$$t=\frac{-30±\sqrt{900-4\times 2\left(-300\right)}}{2\times 2}$$

Multiply -4 times 2.

$$t=\frac{-30±\sqrt{900-8\left(-300\right)}}{2\times 2}$$

Multiply -8 times -300.

$$t=\frac{-30±\sqrt{900+2400}}{2\times 2}$$

Add 900 to 2400.

$$t=\frac{-30±\sqrt{3300}}{2\times 2}$$

Take the square root of 3300.

$$t=\frac{-30±10\sqrt{33}}{2\times 2}$$

Multiply 2 times 2.

$$t=\frac{-30±10\sqrt{33}}{4}$$

Now solve the equation t=\frac{-30±10\sqrt{33}}{4} when ± is plus. Add -30 to 10\sqrt{33}.

$$t=\frac{10\sqrt{33}-30}{4}$$

Divide -30+10\sqrt{33} by 4.

$$t=\frac{5\sqrt{33}-15}{2}$$

Now solve the equation t=\frac{-30±10\sqrt{33}}{4} when ± is minus. Subtract 10\sqrt{33} from -30.

$$t=\frac{-10\sqrt{33}-30}{4}$$

Divide -30-10\sqrt{33} by 4.

$$t=\frac{-5\sqrt{33}-15}{2}$$

The equation is now solved.

$$t=\frac{5\sqrt{33}-15}{2} t=\frac{-5\sqrt{33}-15}{2}$$

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.

$$2t^{2}+30t=300$$

Divide both sides by 2.

$$\frac{2t^{2}+30t}{2}=\frac{300}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$t^{2}+\frac{30}{2}t=\frac{300}{2}$$

Divide 30 by 2.

$$t^{2}+15t=\frac{300}{2}$$

Divide 300 by 2.

$$t^{2}+15t=150$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+15t+\left(\frac{15}{2}\right)^{2}=150+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+15t+\frac{225}{4}=150+\frac{225}{4}$$

Add 150 to \frac{225}{4}.

$$t^{2}+15t+\frac{225}{4}=\frac{825}{4}$$

Factor t^{2}+15t+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{15}{2}\right)^{2}=\frac{825}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{15}{2}\right)^{2}}=\sqrt{\frac{825}{4}}$$

Simplify.

$$t+\frac{15}{2}=\frac{5\sqrt{33}}{2} t+\frac{15}{2}=-\frac{5\sqrt{33}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$t=\frac{5\sqrt{33}-15}{2} t=\frac{-5\sqrt{33}-15}{2}$$