Solve {left(3sqrt{5}right)}^2=25t^2-80t+100

$${ \left(3 \sqrt{ 5 } \right) }^{ 2 } = 25 { t }^{ 2 } -80t+100$$

Solve for t

$$t=1$$

$$t = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

Steps Using the Quadratic Formula

Expand \left(3\sqrt{5}\right)^{2}.

$$3^{2}\left(\sqrt{5}\right)^{2}=25t^{2}-80t+100$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{5}\right)^{2}=25t^{2}-80t+100$$

The square of \sqrt{5} is 5.

$$9\times 5=25t^{2}-80t+100$$

Multiply 9 and 5 to get 45.

$$45=25t^{2}-80t+100$$

Swap sides so that all variable terms are on the left hand side.

$$25t^{2}-80t+100=45$$

Subtract 45 from both sides.

$$25t^{2}-80t+100-45=0$$

Subtract 45 from 100 to get 55.

$$25t^{2}-80t+55=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -80 for b, and 55 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 25\times 55}}{2\times 25}$$

Square -80.

$$t=\frac{-\left(-80\right)±\sqrt{6400-4\times 25\times 55}}{2\times 25}$$

Multiply -4 times 25.

$$t=\frac{-\left(-80\right)±\sqrt{6400-100\times 55}}{2\times 25}$$

Multiply -100 times 55.

$$t=\frac{-\left(-80\right)±\sqrt{6400-5500}}{2\times 25}$$

Add 6400 to -5500.

$$t=\frac{-\left(-80\right)±\sqrt{900}}{2\times 25}$$

Take the square root of 900.

$$t=\frac{-\left(-80\right)±30}{2\times 25}$$

The opposite of -80 is 80.

$$t=\frac{80±30}{2\times 25}$$

Multiply 2 times 25.

$$t=\frac{80±30}{50}$$

Now solve the equation t=\frac{80±30}{50} when ± is plus. Add 80 to 30.

$$t=\frac{110}{50}$$

Reduce the fraction \frac{110}{50} to lowest terms by extracting and canceling out 10.

$$t=\frac{11}{5}$$

Now solve the equation t=\frac{80±30}{50} when ± is minus. Subtract 30 from 80.

$$t=\frac{50}{50}$$

Divide 50 by 50.

$$t=1$$

The equation is now solved.

$$t=\frac{11}{5} t=1$$

Steps for Completing the Square

Expand \left(3\sqrt{5}\right)^{2}.

$$3^{2}\left(\sqrt{5}\right)^{2}=25t^{2}-80t+100$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{5}\right)^{2}=25t^{2}-80t+100$$

The square of \sqrt{5} is 5.

$$9\times 5=25t^{2}-80t+100$$

Multiply 9 and 5 to get 45.

$$45=25t^{2}-80t+100$$

Swap sides so that all variable terms are on the left hand side.

$$25t^{2}-80t+100=45$$

Subtract 100 from both sides.

$$25t^{2}-80t=45-100$$

Subtract 100 from 45 to get -55.

$$25t^{2}-80t=-55$$

Divide both sides by 25.

$$\frac{25t^{2}-80t}{25}=-\frac{55}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$t^{2}+\left(-\frac{80}{25}\right)t=-\frac{55}{25}$$

Reduce the fraction \frac{-80}{25} to lowest terms by extracting and canceling out 5.

$$t^{2}-\frac{16}{5}t=-\frac{55}{25}$$

Reduce the fraction \frac{-55}{25} to lowest terms by extracting and canceling out 5.

$$t^{2}-\frac{16}{5}t=-\frac{11}{5}$$

Divide -\frac{16}{5}, the coefficient of the x term, by 2 to get -\frac{8}{5}. Then add the square of -\frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{16}{5}t+\left(-\frac{8}{5}\right)^{2}=-\frac{11}{5}+\left(-\frac{8}{5}\right)^{2}$$

Square -\frac{8}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{16}{5}t+\frac{64}{25}=-\frac{11}{5}+\frac{64}{25}$$

Add -\frac{11}{5} to \frac{64}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{16}{5}t+\frac{64}{25}=\frac{9}{25}$$

Factor t^{2}-\frac{16}{5}t+\frac{64}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{8}{5}\right)^{2}=\frac{9}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{8}{5}\right)^{2}}=\sqrt{\frac{9}{25}}$$

Simplify.

$$t-\frac{8}{5}=\frac{3}{5} t-\frac{8}{5}=-\frac{3}{5}$$

Add \frac{8}{5} to both sides of the equation.

$$t=\frac{11}{5} t=1$$