Solve 11(1)=-10t^2+44t+30

$$11 ( 1 ) = - 10 t ^ { 2 } + 44 t + 30$$

Solve for t

$$t=\frac{\sqrt{674}}{10}+\frac{11}{5}\approx 4.796150997$$

$$t=-\frac{\sqrt{674}}{10}+\frac{11}{5}\approx -0.396150997$$

Steps Using the Quadratic Formula

Multiply 11 and 1 to get 11.

$$11=-10t^{2}+44t+30$$

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

$$-10t^{2}+44t+30=11$$

Subtract 11 from both sides.

$$-10t^{2}+44t+30-11=0$$

Subtract 11 from 30 to get 19.

$$-10t^{2}+44t+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 44 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-44±\sqrt{44^{2}-4\left(-10\right)\times 19}}{2\left(-10\right)}$$

Square 44.

$$t=\frac{-44±\sqrt{1936-4\left(-10\right)\times 19}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$t=\frac{-44±\sqrt{1936+40\times 19}}{2\left(-10\right)}$$

Multiply 40 times 19.

$$t=\frac{-44±\sqrt{1936+760}}{2\left(-10\right)}$$

Add 1936 to 760.

$$t=\frac{-44±\sqrt{2696}}{2\left(-10\right)}$$

Take the square root of 2696.

$$t=\frac{-44±2\sqrt{674}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$t=\frac{-44±2\sqrt{674}}{-20}$$

Now solve the equation t=\frac{-44±2\sqrt{674}}{-20} when ± is plus. Add -44 to 2\sqrt{674}.

$$t=\frac{2\sqrt{674}-44}{-20}$$

Divide -44+2\sqrt{674} by -20.

$$t=-\frac{\sqrt{674}}{10}+\frac{11}{5}$$

Now solve the equation t=\frac{-44±2\sqrt{674}}{-20} when ± is minus. Subtract 2\sqrt{674} from -44.

$$t=\frac{-2\sqrt{674}-44}{-20}$$

Divide -44-2\sqrt{674} by -20.

$$t=\frac{\sqrt{674}}{10}+\frac{11}{5}$$

The equation is now solved.

$$t=-\frac{\sqrt{674}}{10}+\frac{11}{5} t=\frac{\sqrt{674}}{10}+\frac{11}{5}$$

Steps for Completing the Square

Multiply 11 and 1 to get 11.

$$11=-10t^{2}+44t+30$$

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

$$-10t^{2}+44t+30=11$$

Subtract 30 from both sides.

$$-10t^{2}+44t=11-30$$

Subtract 30 from 11 to get -19.

$$-10t^{2}+44t=-19$$

Divide both sides by -10.

$$\frac{-10t^{2}+44t}{-10}=-\frac{19}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$t^{2}+\frac{44}{-10}t=-\frac{19}{-10}$$

Reduce the fraction \frac{44}{-10} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{22}{5}t=-\frac{19}{-10}$$

Divide -19 by -10.

$$t^{2}-\frac{22}{5}t=\frac{19}{10}$$

Divide -\frac{22}{5}, the coefficient of the x term, by 2 to get -\frac{11}{5}. Then add the square of -\frac{11}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{22}{5}t+\left(-\frac{11}{5}\right)^{2}=\frac{19}{10}+\left(-\frac{11}{5}\right)^{2}$$

Square -\frac{11}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=\frac{19}{10}+\frac{121}{25}$$

Add \frac{19}{10} to \frac{121}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=\frac{337}{50}$$

Factor t^{2}-\frac{22}{5}t+\frac{121}{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}.

$$\left(t-\frac{11}{5}\right)^{2}=\frac{337}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{11}{5}\right)^{2}}=\sqrt{\frac{337}{50}}$$

Simplify.

$$t-\frac{11}{5}=\frac{\sqrt{674}}{10} t-\frac{11}{5}=-\frac{\sqrt{674}}{10}$$

Add \frac{11}{5} to both sides of the equation.

$$t=\frac{\sqrt{674}}{10}+\frac{11}{5} t=-\frac{\sqrt{674}}{10}+\frac{11}{5}$$