Solve 5t^2=2.44t+0.61t^2

$$5 t ^ { 2 } = 2.44 t + 0.61 t ^ { 2 }$$

Solve for t

$$t=\frac{244}{439}\approx 0.555808656$$

$$t=0$$

Steps Using Factoring

Subtract 2.44t from both sides.

$$5t^{2}-2.44t=0.61t^{2}$$

Subtract 0.61t^{2} from both sides.

$$5t^{2}-2.44t-0.61t^{2}=0$$

Combine 5t^{2} and -0.61t^{2} to get 4.39t^{2}.

$$4.39t^{2}-2.44t=0$$

Factor out t.

$$t\left(4.39t-2.44\right)=0$$

To find equation solutions, solve t=0 and \frac{439t}{100}-2.44=0.

$$t=0 t=\frac{244}{439}$$

Steps Using the Quadratic Formula

Subtract 2.44t from both sides.

$$5t^{2}-2.44t=0.61t^{2}$$

Subtract 0.61t^{2} from both sides.

$$5t^{2}-2.44t-0.61t^{2}=0$$

Combine 5t^{2} and -0.61t^{2} to get 4.39t^{2}.

$$4.39t^{2}-2.44t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.39 for a, -2.44 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-2.44\right)±\sqrt{\left(-2.44\right)^{2}}}{2\times 4.39}$$

Take the square root of \left(-2.44\right)^{2}.

$$t=\frac{-\left(-2.44\right)±\frac{61}{25}}{2\times 4.39}$$

The opposite of -2.44 is 2.44.

$$t=\frac{2.44±\frac{61}{25}}{2\times 4.39}$$

Multiply 2 times 4.39.

$$t=\frac{2.44±\frac{61}{25}}{8.78}$$

Now solve the equation t=\frac{2.44±\frac{61}{25}}{8.78} when ± is plus. Add 2.44 to \frac{61}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{122}{25}}{8.78}$$

Divide \frac{122}{25} by 8.78 by multiplying \frac{122}{25} by the reciprocal of 8.78.

$$t=\frac{244}{439}$$

Now solve the equation t=\frac{2.44±\frac{61}{25}}{8.78} when ± is minus. Subtract \frac{61}{25} from 2.44 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{0}{8.78}$$

Divide 0 by 8.78 by multiplying 0 by the reciprocal of 8.78.

$$t=0$$

The equation is now solved.

$$t=\frac{244}{439} t=0$$

Steps for Completing the Square

Subtract 2.44t from both sides.

$$5t^{2}-2.44t=0.61t^{2}$$

Subtract 0.61t^{2} from both sides.

$$5t^{2}-2.44t-0.61t^{2}=0$$

Combine 5t^{2} and -0.61t^{2} to get 4.39t^{2}.

$$4.39t^{2}-2.44t=0$$

Divide both sides of the equation by 4.39, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.39t^{2}-2.44t}{4.39}=\frac{0}{4.39}$$

Dividing by 4.39 undoes the multiplication by 4.39.

$$t^{2}+\left(-\frac{2.44}{4.39}\right)t=\frac{0}{4.39}$$

Divide -2.44 by 4.39 by multiplying -2.44 by the reciprocal of 4.39.

$$t^{2}-\frac{244}{439}t=\frac{0}{4.39}$$

Divide 0 by 4.39 by multiplying 0 by the reciprocal of 4.39.

$$t^{2}-\frac{244}{439}t=0$$

Divide -\frac{244}{439}, the coefficient of the x term, by 2 to get -\frac{122}{439}. Then add the square of -\frac{122}{439} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{244}{439}t+\left(-\frac{122}{439}\right)^{2}=\left(-\frac{122}{439}\right)^{2}$$

Square -\frac{122}{439} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{244}{439}t+\frac{14884}{192721}=\frac{14884}{192721}$$

Factor t^{2}-\frac{244}{439}t+\frac{14884}{192721}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{122}{439}\right)^{2}=\frac{14884}{192721}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{122}{439}\right)^{2}}=\sqrt{\frac{14884}{192721}}$$

Simplify.

$$t-\frac{122}{439}=\frac{122}{439} t-\frac{122}{439}=-\frac{122}{439}$$

Add \frac{122}{439} to both sides of the equation.

$$t=\frac{244}{439} t=0$$