Solve 30t=225(t+10)^2

$$30 t = 225 ( t + 10 ) ^ { 2 }$$

Solve for t

$$t=\frac{-149+\sqrt{299}i}{15}\approx -9.933333333+1.152774431i$$

$$t=\frac{-\sqrt{299}i-149}{15}\approx -9.933333333-1.152774431i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(t+10\right)^{2}.

$$30t=225\left(t^{2}+20t+100\right)$$

Use the distributive property to multiply 225 by t^{2}+20t+100.

$$30t=225t^{2}+4500t+22500$$

Subtract 225t^{2} from both sides.

$$30t-225t^{2}=4500t+22500$$

Subtract 4500t from both sides.

$$30t-225t^{2}-4500t=22500$$

Combine 30t and -4500t to get -4470t.

$$-4470t-225t^{2}=22500$$

Subtract 22500 from both sides.

$$-4470t-225t^{2}-22500=0$$

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.

$$-225t^{2}-4470t-22500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -225 for a, -4470 for b, and -22500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-4470\right)±\sqrt{\left(-4470\right)^{2}-4\left(-225\right)\left(-22500\right)}}{2\left(-225\right)}$$

Square -4470.

$$t=\frac{-\left(-4470\right)±\sqrt{19980900-4\left(-225\right)\left(-22500\right)}}{2\left(-225\right)}$$

Multiply -4 times -225.

$$t=\frac{-\left(-4470\right)±\sqrt{19980900+900\left(-22500\right)}}{2\left(-225\right)}$$

Multiply 900 times -22500.

$$t=\frac{-\left(-4470\right)±\sqrt{19980900-20250000}}{2\left(-225\right)}$$

Add 19980900 to -20250000.

$$t=\frac{-\left(-4470\right)±\sqrt{-269100}}{2\left(-225\right)}$$

Take the square root of -269100.

$$t=\frac{-\left(-4470\right)±30\sqrt{299}i}{2\left(-225\right)}$$

The opposite of -4470 is 4470.

$$t=\frac{4470±30\sqrt{299}i}{2\left(-225\right)}$$

Multiply 2 times -225.

$$t=\frac{4470±30\sqrt{299}i}{-450}$$

Now solve the equation t=\frac{4470±30\sqrt{299}i}{-450} when ± is plus. Add 4470 to 30i\sqrt{299}.

$$t=\frac{4470+30\sqrt{299}i}{-450}$$

Divide 4470+30i\sqrt{299} by -450.

$$t=\frac{-\sqrt{299}i-149}{15}$$

Now solve the equation t=\frac{4470±30\sqrt{299}i}{-450} when ± is minus. Subtract 30i\sqrt{299} from 4470.

$$t=\frac{-30\sqrt{299}i+4470}{-450}$$

Divide 4470-30i\sqrt{299} by -450.

$$t=\frac{-149+\sqrt{299}i}{15}$$

The equation is now solved.

$$t=\frac{-\sqrt{299}i-149}{15} t=\frac{-149+\sqrt{299}i}{15}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(t+10\right)^{2}.

$$30t=225\left(t^{2}+20t+100\right)$$

Use the distributive property to multiply 225 by t^{2}+20t+100.

$$30t=225t^{2}+4500t+22500$$

Subtract 225t^{2} from both sides.

$$30t-225t^{2}=4500t+22500$$

Subtract 4500t from both sides.

$$30t-225t^{2}-4500t=22500$$

Combine 30t and -4500t to get -4470t.

$$-4470t-225t^{2}=22500$$

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.

$$-225t^{2}-4470t=22500$$

Divide both sides by -225.

$$\frac{-225t^{2}-4470t}{-225}=\frac{22500}{-225}$$

Dividing by -225 undoes the multiplication by -225.

$$t^{2}+\left(-\frac{4470}{-225}\right)t=\frac{22500}{-225}$$

Reduce the fraction \frac{-4470}{-225} to lowest terms by extracting and canceling out 15.

$$t^{2}+\frac{298}{15}t=\frac{22500}{-225}$$

Divide 22500 by -225.

$$t^{2}+\frac{298}{15}t=-100$$

Divide \frac{298}{15}, the coefficient of the x term, by 2 to get \frac{149}{15}. Then add the square of \frac{149}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{298}{15}t+\left(\frac{149}{15}\right)^{2}=-100+\left(\frac{149}{15}\right)^{2}$$

Square \frac{149}{15} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{298}{15}t+\frac{22201}{225}=-100+\frac{22201}{225}$$

Add -100 to \frac{22201}{225}.

$$t^{2}+\frac{298}{15}t+\frac{22201}{225}=-\frac{299}{225}$$

Factor t^{2}+\frac{298}{15}t+\frac{22201}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{149}{15}\right)^{2}=-\frac{299}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{149}{15}\right)^{2}}=\sqrt{-\frac{299}{225}}$$

Simplify.

$$t+\frac{149}{15}=\frac{\sqrt{299}i}{15} t+\frac{149}{15}=-\frac{\sqrt{299}i}{15}$$

Subtract \frac{149}{15} from both sides of the equation.

$$t=\frac{-149+\sqrt{299}i}{15} t=\frac{-\sqrt{299}i-149}{15}$$