Solve 1.375=6t-2t^2

$$1.375 = 6 t - 2 t ^ { 2 }$$

Solve for t

$$t=0.25$$

$$t=2.75$$

Steps Using the Quadratic Formula

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

$$6t-2t^{2}=1.375$$

Subtract 1.375 from both sides.

$$6t-2t^{2}-1.375=0$$

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.

$$-2t^{2}+6t-1.375=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 6 for b, and -1.375 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-6±\sqrt{6^{2}-4\left(-2\right)\left(-1.375\right)}}{2\left(-2\right)}$$

Square 6.

$$t=\frac{-6±\sqrt{36-4\left(-2\right)\left(-1.375\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$t=\frac{-6±\sqrt{36+8\left(-1.375\right)}}{2\left(-2\right)}$$

Multiply 8 times -1.375.

$$t=\frac{-6±\sqrt{36-11}}{2\left(-2\right)}$$

Add 36 to -11.

$$t=\frac{-6±\sqrt{25}}{2\left(-2\right)}$$

Take the square root of 25.

$$t=\frac{-6±5}{2\left(-2\right)}$$

Multiply 2 times -2.

$$t=\frac{-6±5}{-4}$$

Now solve the equation t=\frac{-6±5}{-4} when ± is plus. Add -6 to 5.

$$t=-\frac{1}{-4}$$

Divide -1 by -4.

$$t=\frac{1}{4}$$

Now solve the equation t=\frac{-6±5}{-4} when ± is minus. Subtract 5 from -6.

$$t=-\frac{11}{-4}$$

Divide -11 by -4.

$$t=\frac{11}{4}$$

The equation is now solved.

$$t=\frac{1}{4} t=\frac{11}{4}$$

Steps for Completing the Square

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

$$6t-2t^{2}=1.375$$

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.

$$-2t^{2}+6t=1.375$$

Divide both sides by -2.

$$\frac{-2t^{2}+6t}{-2}=\frac{1.375}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$t^{2}+\frac{6}{-2}t=\frac{1.375}{-2}$$

Divide 6 by -2.

$$t^{2}-3t=\frac{1.375}{-2}$$

Divide 1.375 by -2.

$$t^{2}-3t=-0.6875$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-3t+\left(-\frac{3}{2}\right)^{2}=-0.6875+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-3t+\frac{9}{4}=-0.6875+\frac{9}{4}$$

Add -0.6875 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-3t+\frac{9}{4}=\frac{25}{16}$$

Factor t^{2}-3t+\frac{9}{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}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$t-\frac{3}{2}=\frac{5}{4} t-\frac{3}{2}=-\frac{5}{4}$$

Add \frac{3}{2} to both sides of the equation.

$$t=\frac{11}{4} t=\frac{1}{4}$$