Solve [10-5t)t=9.375

$$[ 10 - 5 t ) t = 9.375$$

Solve for t

$$t=\frac{i\sqrt{14}}{4}+1\approx 1+0.935414347i$$

$$t=-\frac{i\sqrt{14}}{4}+1\approx 1-0.935414347i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10-5t by t.

$$10t-5t^{2}=9.375$$

Subtract 9.375 from both sides.

$$10t-5t^{2}-9.375=0$$

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.

$$-5t^{2}+10t-9.375=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 10 for b, and -9.375 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\left(-9.375\right)}}{2\left(-5\right)}$$

Square 10.

$$t=\frac{-10±\sqrt{100-4\left(-5\right)\left(-9.375\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$t=\frac{-10±\sqrt{100+20\left(-9.375\right)}}{2\left(-5\right)}$$

Multiply 20 times -9.375.

$$t=\frac{-10±\sqrt{100-187.5}}{2\left(-5\right)}$$

Add 100 to -187.5.

$$t=\frac{-10±\sqrt{-87.5}}{2\left(-5\right)}$$

Take the square root of -87.5.

$$t=\frac{-10±\frac{5\sqrt{14}i}{2}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$t=\frac{-10±\frac{5\sqrt{14}i}{2}}{-10}$$

Now solve the equation t=\frac{-10±\frac{5\sqrt{14}i}{2}}{-10} when ± is plus. Add -10 to \frac{5i\sqrt{14}}{2}.

$$t=\frac{\frac{5\sqrt{14}i}{2}-10}{-10}$$

Divide -10+\frac{5i\sqrt{14}}{2} by -10.

$$t=-\frac{\sqrt{14}i}{4}+1$$

Now solve the equation t=\frac{-10±\frac{5\sqrt{14}i}{2}}{-10} when ± is minus. Subtract \frac{5i\sqrt{14}}{2} from -10.

$$t=\frac{-\frac{5\sqrt{14}i}{2}-10}{-10}$$

Divide -10-\frac{5i\sqrt{14}}{2} by -10.

$$t=\frac{\sqrt{14}i}{4}+1$$

The equation is now solved.

$$t=-\frac{\sqrt{14}i}{4}+1 t=\frac{\sqrt{14}i}{4}+1$$

Steps for Completing the Square

Use the distributive property to multiply 10-5t by t.

$$10t-5t^{2}=9.375$$

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.

$$-5t^{2}+10t=9.375$$

Divide both sides by -5.

$$\frac{-5t^{2}+10t}{-5}=\frac{9.375}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$t^{2}+\frac{10}{-5}t=\frac{9.375}{-5}$$

Divide 10 by -5.

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

Divide 9.375 by -5.

$$t^{2}-2t=-1.875$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-2t+1=-1.875+1$$

Add -1.875 to 1.

$$t^{2}-2t+1=-0.875$$

Factor t^{2}-2t+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-1\right)^{2}=-0.875$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-1\right)^{2}}=\sqrt{-0.875}$$

Simplify.

$$t-1=\frac{\sqrt{14}i}{4} t-1=-\frac{\sqrt{14}i}{4}$$

Add 1 to both sides of the equation.

$$t=\frac{\sqrt{14}i}{4}+1 t=-\frac{\sqrt{14}i}{4}+1$$