Solve left(10-2tright)t=9.375

$$\left( 10-2t \right) t = 9.375$$

Solve for t

$$t=1.25$$

$$t=3.75$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10-2t by t.

$$10t-2t^{2}=9.375$$

Subtract 9.375 from both sides.

$$10t-2t^{2}-9.375=0$$

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.

$$-2t^{2}+10t-9.375=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and -9.375 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-9.375\right)}}{2\left(-2\right)}$$

Square 10.

$$t=\frac{-10±\sqrt{100-4\left(-2\right)\left(-9.375\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$t=\frac{-10±\sqrt{100+8\left(-9.375\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.375.

$$t=\frac{-10±\sqrt{100-75}}{2\left(-2\right)}$$

Add 100 to -75.

$$t=\frac{-10±\sqrt{25}}{2\left(-2\right)}$$

Take the square root of 25.

$$t=\frac{-10±5}{2\left(-2\right)}$$

Multiply 2 times -2.

$$t=\frac{-10±5}{-4}$$

Now solve the equation t=\frac{-10±5}{-4} when ± is plus. Add -10 to 5.

$$t=-\frac{5}{-4}$$

Divide -5 by -4.

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

Now solve the equation t=\frac{-10±5}{-4} when ± is minus. Subtract 5 from -10.

$$t=-\frac{15}{-4}$$

Divide -15 by -4.

$$t=\frac{15}{4}$$

The equation is now solved.

$$t=\frac{5}{4} t=\frac{15}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 10-2t by t.

$$10t-2t^{2}=9.375$$

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.

$$-2t^{2}+10t=9.375$$

Divide both sides by -2.

$$\frac{-2t^{2}+10t}{-2}=\frac{9.375}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$t^{2}+\frac{10}{-2}t=\frac{9.375}{-2}$$

Divide 10 by -2.

$$t^{2}-5t=\frac{9.375}{-2}$$

Divide 9.375 by -2.

$$t^{2}-5t=-4.6875$$

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

$$t^{2}-5t+\left(-\frac{5}{2}\right)^{2}=-4.6875+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-5t+\frac{25}{4}=-4.6875+\frac{25}{4}$$

Add -4.6875 to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-5t+\frac{25}{4}=\frac{25}{16}$$

Factor t^{2}-5t+\frac{25}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$t-\frac{5}{2}=\frac{5}{4} t-\frac{5}{2}=-\frac{5}{4}$$

Add \frac{5}{2} to both sides of the equation.

$$t=\frac{15}{4} t=\frac{5}{4}$$