Solve 50=1.9t+4.3t^2

$$50 = 1.9 t + 4.3 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{86361} - 19}{86} \approx 3.196190945$$

$$t=\frac{-\sqrt{86361}-19}{86}\approx -3.63805141$$

Steps Using the Quadratic Formula

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

$$1.9t+4.3t^{2}=50$$

Subtract 50 from both sides.

$$1.9t+4.3t^{2}-50=0$$

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.

$$4.3t^{2}+1.9t-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.3 for a, 1.9 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-1.9±\sqrt{1.9^{2}-4\times 4.3\left(-50\right)}}{2\times 4.3}$$

Square 1.9 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-1.9±\sqrt{3.61-4\times 4.3\left(-50\right)}}{2\times 4.3}$$

Multiply -4 times 4.3.

$$t=\frac{-1.9±\sqrt{3.61-17.2\left(-50\right)}}{2\times 4.3}$$

Multiply -17.2 times -50.

$$t=\frac{-1.9±\sqrt{3.61+860}}{2\times 4.3}$$

Add 3.61 to 860.

$$t=\frac{-1.9±\sqrt{863.61}}{2\times 4.3}$$

Take the square root of 863.61.

$$t=\frac{-1.9±\frac{\sqrt{86361}}{10}}{2\times 4.3}$$

Multiply 2 times 4.3.

$$t=\frac{-1.9±\frac{\sqrt{86361}}{10}}{8.6}$$

Now solve the equation t=\frac{-1.9±\frac{\sqrt{86361}}{10}}{8.6} when ± is plus. Add -1.9 to \frac{\sqrt{86361}}{10}.

$$t=\frac{\sqrt{86361}-19}{8.6\times 10}$$

Divide \frac{-19+\sqrt{86361}}{10} by 8.6 by multiplying \frac{-19+\sqrt{86361}}{10} by the reciprocal of 8.6.

$$t=\frac{\sqrt{86361}-19}{86}$$

Now solve the equation t=\frac{-1.9±\frac{\sqrt{86361}}{10}}{8.6} when ± is minus. Subtract \frac{\sqrt{86361}}{10} from -1.9.

$$t=\frac{-\sqrt{86361}-19}{8.6\times 10}$$

Divide \frac{-19-\sqrt{86361}}{10} by 8.6 by multiplying \frac{-19-\sqrt{86361}}{10} by the reciprocal of 8.6.

$$t=\frac{-\sqrt{86361}-19}{86}$$

The equation is now solved.

$$t=\frac{\sqrt{86361}-19}{86} t=\frac{-\sqrt{86361}-19}{86}$$

Steps for Completing the Square

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

$$1.9t+4.3t^{2}=50$$

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.

$$4.3t^{2}+1.9t=50$$

Divide both sides of the equation by 4.3, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.3t^{2}+1.9t}{4.3}=\frac{50}{4.3}$$

Dividing by 4.3 undoes the multiplication by 4.3.

$$t^{2}+\frac{1.9}{4.3}t=\frac{50}{4.3}$$

Divide 1.9 by 4.3 by multiplying 1.9 by the reciprocal of 4.3.

$$t^{2}+\frac{19}{43}t=\frac{50}{4.3}$$

Divide 50 by 4.3 by multiplying 50 by the reciprocal of 4.3.

$$t^{2}+\frac{19}{43}t=\frac{500}{43}$$

Divide \frac{19}{43}, the coefficient of the x term, by 2 to get \frac{19}{86}. Then add the square of \frac{19}{86} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{19}{43}t+\frac{19}{86}^{2}=\frac{500}{43}+\frac{19}{86}^{2}$$

Square \frac{19}{86} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{19}{43}t+\frac{361}{7396}=\frac{500}{43}+\frac{361}{7396}$$

Add \frac{500}{43} to \frac{361}{7396} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{19}{43}t+\frac{361}{7396}=\frac{86361}{7396}$$

Factor t^{2}+\frac{19}{43}t+\frac{361}{7396}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{19}{86}\right)^{2}=\frac{86361}{7396}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{19}{86}\right)^{2}}=\sqrt{\frac{86361}{7396}}$$

Simplify.

$$t+\frac{19}{86}=\frac{\sqrt{86361}}{86} t+\frac{19}{86}=-\frac{\sqrt{86361}}{86}$$

Subtract \frac{19}{86} from both sides of the equation.

$$t=\frac{\sqrt{86361}-19}{86} t=\frac{-\sqrt{86361}-19}{86}$$