Solve 5(4,5-t)^2=320t

$$5 ( 4,5 - t ) ^ { 2 } = 320 t$$

Solve for t

$$t=4\sqrt{82}+36,5\approx 72.721540553$$

$$t=36,5-4\sqrt{82}\approx 0.278459447$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4,5-t\right)^{2}.

$$5\left(20,25-9t+t^{2}\right)=320t$$

Use the distributive property to multiply 5 by 20,25-9t+t^{2}.

$$101,25-45t+5t^{2}=320t$$

Subtract 320t from both sides.

$$101,25-45t+5t^{2}-320t=0$$

Combine -45t and -320t to get -365t.

$$101,25-365t+5t^{2}=0$$

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.

$$5t^{2}-365t+101,25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -365 for b, and 101,25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-365\right)±\sqrt{\left(-365\right)^{2}-4\times 5\times 101,25}}{2\times 5}$$

Square -365.

$$t=\frac{-\left(-365\right)±\sqrt{133225-4\times 5\times 101,25}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-\left(-365\right)±\sqrt{133225-20\times 101,25}}{2\times 5}$$

Multiply -20 times 101,25.

$$t=\frac{-\left(-365\right)±\sqrt{133225-2025}}{2\times 5}$$

Add 133225 to -2025.

$$t=\frac{-\left(-365\right)±\sqrt{131200}}{2\times 5}$$

Take the square root of 131200.

$$t=\frac{-\left(-365\right)±40\sqrt{82}}{2\times 5}$$

The opposite of -365 is 365.

$$t=\frac{365±40\sqrt{82}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{365±40\sqrt{82}}{10}$$

Now solve the equation t=\frac{365±40\sqrt{82}}{10} when ± is plus. Add 365 to 40\sqrt{82}.

$$t=\frac{40\sqrt{82}+365}{10}$$

Divide 365+40\sqrt{82} by 10.

$$t=4\sqrt{82}+\frac{73}{2}$$

Now solve the equation t=\frac{365±40\sqrt{82}}{10} when ± is minus. Subtract 40\sqrt{82} from 365.

$$t=\frac{365-40\sqrt{82}}{10}$$

Divide 365-40\sqrt{82} by 10.

$$t=\frac{73}{2}-4\sqrt{82}$$

The equation is now solved.

$$t=4\sqrt{82}+\frac{73}{2} t=\frac{73}{2}-4\sqrt{82}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4,5-t\right)^{2}.

$$5\left(20,25-9t+t^{2}\right)=320t$$

Use the distributive property to multiply 5 by 20,25-9t+t^{2}.

$$101,25-45t+5t^{2}=320t$$

Subtract 320t from both sides.

$$101,25-45t+5t^{2}-320t=0$$

Combine -45t and -320t to get -365t.

$$101,25-365t+5t^{2}=0$$

Subtract 101,25 from both sides. Anything subtracted from zero gives its negation.

$$-365t+5t^{2}=-101,25$$

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.

$$5t^{2}-365t=-101,25$$

Divide both sides by 5.

$$\frac{5t^{2}-365t}{5}=-\frac{101,25}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\left(-\frac{365}{5}\right)t=-\frac{101,25}{5}$$

Divide -365 by 5.

$$t^{2}-73t=-\frac{101,25}{5}$$

Divide -101,25 by 5.

$$t^{2}-73t=-20,25$$

Divide -73, the coefficient of the x term, by 2 to get -\frac{73}{2}. Then add the square of -\frac{73}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-73t+\left(-\frac{73}{2}\right)^{2}=-20,25+\left(-\frac{73}{2}\right)^{2}$$

Square -\frac{73}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-73t+\frac{5329}{4}=\frac{-81+5329}{4}$$

Add -20,25 to \frac{5329}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-73t+\frac{5329}{4}=1312$$

Factor t^{2}-73t+\frac{5329}{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}.

$$\left(t-\frac{73}{2}\right)^{2}=1312$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{73}{2}\right)^{2}}=\sqrt{1312}$$

Simplify.

$$t-\frac{73}{2}=4\sqrt{82} t-\frac{73}{2}=-4\sqrt{82}$$

Add \frac{73}{2} to both sides of the equation.

$$t=4\sqrt{82}+\frac{73}{2} t=\frac{73}{2}-4\sqrt{82}$$