Solve 76=70t-16t^2

$$76 = 70t-16 { t }^{ 2 }$$

Solve for t

$$t=2$$

$$t = \frac{19}{8} = 2\frac{3}{8} = 2.375$$

Steps Using the Quadratic Formula

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

$$70t-16t^{2}=76$$

Subtract 76 from both sides.

$$70t-16t^{2}-76=0$$

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.

$$-16t^{2}+70t-76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 70 for b, and -76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-70±\sqrt{70^{2}-4\left(-16\right)\left(-76\right)}}{2\left(-16\right)}$$

Square 70.

$$t=\frac{-70±\sqrt{4900-4\left(-16\right)\left(-76\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-70±\sqrt{4900+64\left(-76\right)}}{2\left(-16\right)}$$

Multiply 64 times -76.

$$t=\frac{-70±\sqrt{4900-4864}}{2\left(-16\right)}$$

Add 4900 to -4864.

$$t=\frac{-70±\sqrt{36}}{2\left(-16\right)}$$

Take the square root of 36.

$$t=\frac{-70±6}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-70±6}{-32}$$

Now solve the equation t=\frac{-70±6}{-32} when ± is plus. Add -70 to 6.

$$t=-\frac{64}{-32}$$

Divide -64 by -32.

$$t=2$$

Now solve the equation t=\frac{-70±6}{-32} when ± is minus. Subtract 6 from -70.

$$t=-\frac{76}{-32}$$

Reduce the fraction \frac{-76}{-32} to lowest terms by extracting and canceling out 4.

$$t=\frac{19}{8}$$

The equation is now solved.

$$t=2 t=\frac{19}{8}$$

Steps for Completing the Square

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

$$70t-16t^{2}=76$$

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.

$$-16t^{2}+70t=76$$

Divide both sides by -16.

$$\frac{-16t^{2}+70t}{-16}=\frac{76}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{70}{-16}t=\frac{76}{-16}$$

Reduce the fraction \frac{70}{-16} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{35}{8}t=\frac{76}{-16}$$

Reduce the fraction \frac{76}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{35}{8}t=-\frac{19}{4}$$

Divide -\frac{35}{8}, the coefficient of the x term, by 2 to get -\frac{35}{16}. Then add the square of -\frac{35}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{35}{8}t+\left(-\frac{35}{16}\right)^{2}=-\frac{19}{4}+\left(-\frac{35}{16}\right)^{2}$$

Square -\frac{35}{16} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{35}{8}t+\frac{1225}{256}=-\frac{19}{4}+\frac{1225}{256}$$

Add -\frac{19}{4} to \frac{1225}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{35}{8}t+\frac{1225}{256}=\frac{9}{256}$$

Factor t^{2}-\frac{35}{8}t+\frac{1225}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{35}{16}\right)^{2}=\frac{9}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{35}{16}\right)^{2}}=\sqrt{\frac{9}{256}}$$

Simplify.

$$t-\frac{35}{16}=\frac{3}{16} t-\frac{35}{16}=-\frac{3}{16}$$

Add \frac{35}{16} to both sides of the equation.

$$t=\frac{19}{8} t=2$$