Solve 4-4t=2.25t^2

$$4 - 4 t = 2.25 t ^ { 2 }$$

Solve for t

$$t=\frac{4\sqrt{13}-8}{9}\approx 0.713578345$$

$$t=\frac{-4\sqrt{13}-8}{9}\approx -2.491356122$$

Steps Using the Quadratic Formula

Subtract 2.25t^{2} from both sides.

$$4-4t-2.25t^{2}=0$$

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.

$$-2.25t^{2}-4t+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2.25 for a, -4 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2.25\right)\times 4}}{2\left(-2.25\right)}$$

Square -4.

$$t=\frac{-\left(-4\right)±\sqrt{16-4\left(-2.25\right)\times 4}}{2\left(-2.25\right)}$$

Multiply -4 times -2.25.

$$t=\frac{-\left(-4\right)±\sqrt{16+9\times 4}}{2\left(-2.25\right)}$$

Multiply 9 times 4.

$$t=\frac{-\left(-4\right)±\sqrt{16+36}}{2\left(-2.25\right)}$$

Add 16 to 36.

$$t=\frac{-\left(-4\right)±\sqrt{52}}{2\left(-2.25\right)}$$

Take the square root of 52.

$$t=\frac{-\left(-4\right)±2\sqrt{13}}{2\left(-2.25\right)}$$

The opposite of -4 is 4.

$$t=\frac{4±2\sqrt{13}}{2\left(-2.25\right)}$$

Multiply 2 times -2.25.

$$t=\frac{4±2\sqrt{13}}{-4.5}$$

Now solve the equation t=\frac{4±2\sqrt{13}}{-4.5} when ± is plus. Add 4 to 2\sqrt{13}.

$$t=\frac{2\sqrt{13}+4}{-4.5}$$

Divide 4+2\sqrt{13} by -4.5 by multiplying 4+2\sqrt{13} by the reciprocal of -4.5.

$$t=\frac{-4\sqrt{13}-8}{9}$$

Now solve the equation t=\frac{4±2\sqrt{13}}{-4.5} when ± is minus. Subtract 2\sqrt{13} from 4.

$$t=\frac{4-2\sqrt{13}}{-4.5}$$

Divide 4-2\sqrt{13} by -4.5 by multiplying 4-2\sqrt{13} by the reciprocal of -4.5.

$$t=\frac{4\sqrt{13}-8}{9}$$

The equation is now solved.

$$t=\frac{-4\sqrt{13}-8}{9} t=\frac{4\sqrt{13}-8}{9}$$

Steps for Completing the Square

Subtract 2.25t^{2} from both sides.

$$4-4t-2.25t^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-4t-2.25t^{2}=-4$$

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.

$$-2.25t^{2}-4t=-4$$

Divide both sides of the equation by -2.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-2.25t^{2}-4t}{-2.25}=-\frac{4}{-2.25}$$

Dividing by -2.25 undoes the multiplication by -2.25.

$$t^{2}+\left(-\frac{4}{-2.25}\right)t=-\frac{4}{-2.25}$$

Divide -4 by -2.25 by multiplying -4 by the reciprocal of -2.25.

$$t^{2}+\frac{16}{9}t=-\frac{4}{-2.25}$$

Divide -4 by -2.25 by multiplying -4 by the reciprocal of -2.25.

$$t^{2}+\frac{16}{9}t=\frac{16}{9}$$

Divide \frac{16}{9}, the coefficient of the x term, by 2 to get \frac{8}{9}. Then add the square of \frac{8}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{16}{9}t+\frac{8}{9}^{2}=\frac{16}{9}+\frac{8}{9}^{2}$$

Square \frac{8}{9} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{16}{9}t+\frac{64}{81}=\frac{16}{9}+\frac{64}{81}$$

Add \frac{16}{9} to \frac{64}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{16}{9}t+\frac{64}{81}=\frac{208}{81}$$

Factor t^{2}+\frac{16}{9}t+\frac{64}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{8}{9}\right)^{2}=\frac{208}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{8}{9}\right)^{2}}=\sqrt{\frac{208}{81}}$$

Simplify.

$$t+\frac{8}{9}=\frac{4\sqrt{13}}{9} t+\frac{8}{9}=-\frac{4\sqrt{13}}{9}$$

Subtract \frac{8}{9} from both sides of the equation.

$$t=\frac{4\sqrt{13}-8}{9} t=\frac{-4\sqrt{13}-8}{9}$$