Solve [30+(30+4t)t]=600

$$[ 30 + ( 30 + 4 t ) t ] = 600$$

Solve for t

$$t = \frac{\sqrt{2505} - 15}{4} \approx 8.762493756$$

$$t=\frac{-\sqrt{2505}-15}{4}\approx -16.262493756$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 30+4t by t.

$$30+30t+4t^{2}=600$$

Subtract 600 from both sides.

$$30+30t+4t^{2}-600=0$$

Subtract 600 from 30 to get -570.

$$-570+30t+4t^{2}=0$$

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.

$$4t^{2}+30t-570=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 30 for b, and -570 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-30±\sqrt{30^{2}-4\times 4\left(-570\right)}}{2\times 4}$$

Square 30.

$$t=\frac{-30±\sqrt{900-4\times 4\left(-570\right)}}{2\times 4}$$

Multiply -4 times 4.

$$t=\frac{-30±\sqrt{900-16\left(-570\right)}}{2\times 4}$$

Multiply -16 times -570.

$$t=\frac{-30±\sqrt{900+9120}}{2\times 4}$$

Add 900 to 9120.

$$t=\frac{-30±\sqrt{10020}}{2\times 4}$$

Take the square root of 10020.

$$t=\frac{-30±2\sqrt{2505}}{2\times 4}$$

Multiply 2 times 4.

$$t=\frac{-30±2\sqrt{2505}}{8}$$

Now solve the equation t=\frac{-30±2\sqrt{2505}}{8} when ± is plus. Add -30 to 2\sqrt{2505}.

$$t=\frac{2\sqrt{2505}-30}{8}$$

Divide -30+2\sqrt{2505} by 8.

$$t=\frac{\sqrt{2505}-15}{4}$$

Now solve the equation t=\frac{-30±2\sqrt{2505}}{8} when ± is minus. Subtract 2\sqrt{2505} from -30.

$$t=\frac{-2\sqrt{2505}-30}{8}$$

Divide -30-2\sqrt{2505} by 8.

$$t=\frac{-\sqrt{2505}-15}{4}$$

The equation is now solved.

$$t=\frac{\sqrt{2505}-15}{4} t=\frac{-\sqrt{2505}-15}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 30+4t by t.

$$30+30t+4t^{2}=600$$

Subtract 30 from both sides.

$$30t+4t^{2}=600-30$$

Subtract 30 from 600 to get 570.

$$30t+4t^{2}=570$$

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.

$$4t^{2}+30t=570$$

Divide both sides by 4.

$$\frac{4t^{2}+30t}{4}=\frac{570}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$t^{2}+\frac{30}{4}t=\frac{570}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$t^{2}+\frac{15}{2}t=\frac{570}{4}$$

Reduce the fraction \frac{570}{4} to lowest terms by extracting and canceling out 2.

$$t^{2}+\frac{15}{2}t=\frac{285}{2}$$

Divide \frac{15}{2}, the coefficient of the x term, by 2 to get \frac{15}{4}. Then add the square of \frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{15}{2}t+\left(\frac{15}{4}\right)^{2}=\frac{285}{2}+\left(\frac{15}{4}\right)^{2}$$

Square \frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{15}{2}t+\frac{225}{16}=\frac{285}{2}+\frac{225}{16}$$

Add \frac{285}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{15}{2}t+\frac{225}{16}=\frac{2505}{16}$$

Factor t^{2}+\frac{15}{2}t+\frac{225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{15}{4}\right)^{2}=\frac{2505}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{15}{4}\right)^{2}}=\sqrt{\frac{2505}{16}}$$

Simplify.

$$t+\frac{15}{4}=\frac{\sqrt{2505}}{4} t+\frac{15}{4}=-\frac{\sqrt{2505}}{4}$$

Subtract \frac{15}{4} from both sides of the equation.

$$t=\frac{\sqrt{2505}-15}{4} t=\frac{-\sqrt{2505}-15}{4}$$