Solve (16-5t)^2+6^2=10^2

$$( 16 - 5 t ) ^ { 2 } + 6 ^ { 2 } = 10 ^ { 2 }$$

Solve for t

$$t = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

$$t = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16-5t\right)^{2}.

$$256-160t+25t^{2}+6^{2}=10^{2}$$

Calculate 6 to the power of 2 and get 36.

$$256-160t+25t^{2}+36=10^{2}$$

Add 256 and 36 to get 292.

$$292-160t+25t^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$292-160t+25t^{2}=100$$

Subtract 100 from both sides.

$$292-160t+25t^{2}-100=0$$

Subtract 100 from 292 to get 192.

$$192-160t+25t^{2}=0$$

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.

$$25t^{2}-160t+192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -160 for b, and 192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-160\right)±\sqrt{\left(-160\right)^{2}-4\times 25\times 192}}{2\times 25}$$

Square -160.

$$t=\frac{-\left(-160\right)±\sqrt{25600-4\times 25\times 192}}{2\times 25}$$

Multiply -4 times 25.

$$t=\frac{-\left(-160\right)±\sqrt{25600-100\times 192}}{2\times 25}$$

Multiply -100 times 192.

$$t=\frac{-\left(-160\right)±\sqrt{25600-19200}}{2\times 25}$$

Add 25600 to -19200.

$$t=\frac{-\left(-160\right)±\sqrt{6400}}{2\times 25}$$

Take the square root of 6400.

$$t=\frac{-\left(-160\right)±80}{2\times 25}$$

The opposite of -160 is 160.

$$t=\frac{160±80}{2\times 25}$$

Multiply 2 times 25.

$$t=\frac{160±80}{50}$$

Now solve the equation t=\frac{160±80}{50} when ± is plus. Add 160 to 80.

$$t=\frac{240}{50}$$

Reduce the fraction \frac{240}{50} to lowest terms by extracting and canceling out 10.

$$t=\frac{24}{5}$$

Now solve the equation t=\frac{160±80}{50} when ± is minus. Subtract 80 from 160.

$$t=\frac{80}{50}$$

Reduce the fraction \frac{80}{50} to lowest terms by extracting and canceling out 10.

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

The equation is now solved.

$$t=\frac{24}{5} t=\frac{8}{5}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16-5t\right)^{2}.

$$256-160t+25t^{2}+6^{2}=10^{2}$$

Calculate 6 to the power of 2 and get 36.

$$256-160t+25t^{2}+36=10^{2}$$

Add 256 and 36 to get 292.

$$292-160t+25t^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$292-160t+25t^{2}=100$$

Subtract 292 from both sides.

$$-160t+25t^{2}=100-292$$

Subtract 292 from 100 to get -192.

$$-160t+25t^{2}=-192$$

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.

$$25t^{2}-160t=-192$$

Divide both sides by 25.

$$\frac{25t^{2}-160t}{25}=-\frac{192}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$t^{2}+\left(-\frac{160}{25}\right)t=-\frac{192}{25}$$

Reduce the fraction \frac{-160}{25} to lowest terms by extracting and canceling out 5.

$$t^{2}-\frac{32}{5}t=-\frac{192}{25}$$

Divide -\frac{32}{5}, the coefficient of the x term, by 2 to get -\frac{16}{5}. Then add the square of -\frac{16}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{32}{5}t+\left(-\frac{16}{5}\right)^{2}=-\frac{192}{25}+\left(-\frac{16}{5}\right)^{2}$$

Square -\frac{16}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{32}{5}t+\frac{256}{25}=\frac{-192+256}{25}$$

Add -\frac{192}{25} to \frac{256}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{32}{5}t+\frac{256}{25}=\frac{64}{25}$$

Factor t^{2}-\frac{32}{5}t+\frac{256}{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}.

$$\left(t-\frac{16}{5}\right)^{2}=\frac{64}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{16}{5}\right)^{2}}=\sqrt{\frac{64}{25}}$$

Simplify.

$$t-\frac{16}{5}=\frac{8}{5} t-\frac{16}{5}=-\frac{8}{5}$$

Add \frac{16}{5} to both sides of the equation.

$$t=\frac{24}{5} t=\frac{8}{5}$$