Solve -5.2=6t-5t^2

$$-5.2 = 6t-5 { t }^{ 2 }$$

Solve for t

$$t = \frac{\sqrt{35} + 3}{5} \approx 1.783215957$$

$$t=\frac{3-\sqrt{35}}{5}\approx -0.583215957$$

Steps Using the Quadratic Formula

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

$$6t-5t^{2}=-5.2$$

Add 5.2 to both sides.

$$6t-5t^{2}+5.2=0$$

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.

$$-5t^{2}+6t+5.2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 6 for b, and 5.2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-6±\sqrt{6^{2}-4\left(-5\right)\times 5.2}}{2\left(-5\right)}$$

Square 6.

$$t=\frac{-6±\sqrt{36-4\left(-5\right)\times 5.2}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$t=\frac{-6±\sqrt{36+20\times 5.2}}{2\left(-5\right)}$$

Multiply 20 times 5.2.

$$t=\frac{-6±\sqrt{36+104}}{2\left(-5\right)}$$

Add 36 to 104.

$$t=\frac{-6±\sqrt{140}}{2\left(-5\right)}$$

Take the square root of 140.

$$t=\frac{-6±2\sqrt{35}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$t=\frac{-6±2\sqrt{35}}{-10}$$

Now solve the equation t=\frac{-6±2\sqrt{35}}{-10} when ± is plus. Add -6 to 2\sqrt{35}.

$$t=\frac{2\sqrt{35}-6}{-10}$$

Divide -6+2\sqrt{35} by -10.

$$t=\frac{3-\sqrt{35}}{5}$$

Now solve the equation t=\frac{-6±2\sqrt{35}}{-10} when ± is minus. Subtract 2\sqrt{35} from -6.

$$t=\frac{-2\sqrt{35}-6}{-10}$$

Divide -6-2\sqrt{35} by -10.

$$t=\frac{\sqrt{35}+3}{5}$$

The equation is now solved.

$$t=\frac{3-\sqrt{35}}{5} t=\frac{\sqrt{35}+3}{5}$$

Steps for Completing the Square

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

$$6t-5t^{2}=-5.2$$

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.

$$-5t^{2}+6t=-5.2$$

Divide both sides by -5.

$$\frac{-5t^{2}+6t}{-5}=-\frac{5.2}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$t^{2}+\frac{6}{-5}t=-\frac{5.2}{-5}$$

Divide 6 by -5.

$$t^{2}-\frac{6}{5}t=-\frac{5.2}{-5}$$

Divide -5.2 by -5.

$$t^{2}-\frac{6}{5}t=1.04$$

Divide -\frac{6}{5}, the coefficient of the x term, by 2 to get -\frac{3}{5}. Then add the square of -\frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{6}{5}t+\left(-\frac{3}{5}\right)^{2}=1.04+\left(-\frac{3}{5}\right)^{2}$$

Square -\frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{6}{5}t+\frac{9}{25}=\frac{26+9}{25}$$

Add 1.04 to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{6}{5}t+\frac{9}{25}=\frac{7}{5}$$

Factor t^{2}-\frac{6}{5}t+\frac{9}{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}.

$$\left(t-\frac{3}{5}\right)^{2}=\frac{7}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{5}\right)^{2}}=\sqrt{\frac{7}{5}}$$

Simplify.

$$t-\frac{3}{5}=\frac{\sqrt{35}}{5} t-\frac{3}{5}=-\frac{\sqrt{35}}{5}$$

Add \frac{3}{5} to both sides of the equation.

$$t=\frac{\sqrt{35}+3}{5} t=\frac{3-\sqrt{35}}{5}$$