Solve 24=t+0.25t^2

$$24 = t + 0.25 t ^ { 2 }$$

Solve for t

$$t=-12$$

$$t=8$$

Steps Using the Quadratic Formula

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

$$t+0.25t^{2}=24$$

Subtract 24 from both sides.

$$t+0.25t^{2}-24=0$$

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.

$$0.25t^{2}+t-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.25 for a, 1 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-1±\sqrt{1^{2}-4\times 0.25\left(-24\right)}}{2\times 0.25}$$

Square 1.

$$t=\frac{-1±\sqrt{1-4\times 0.25\left(-24\right)}}{2\times 0.25}$$

Multiply -4 times 0.25.

$$t=\frac{-1±\sqrt{1-\left(-24\right)}}{2\times 0.25}$$

Multiply -1 times -24.

$$t=\frac{-1±\sqrt{1+24}}{2\times 0.25}$$

Add 1 to 24.

$$t=\frac{-1±\sqrt{25}}{2\times 0.25}$$

Take the square root of 25.

$$t=\frac{-1±5}{2\times 0.25}$$

Multiply 2 times 0.25.

$$t=\frac{-1±5}{0.5}$$

Now solve the equation t=\frac{-1±5}{0.5} when ± is plus. Add -1 to 5.

$$t=\frac{4}{0.5}$$

Divide 4 by 0.5 by multiplying 4 by the reciprocal of 0.5.

$$t=8$$

Now solve the equation t=\frac{-1±5}{0.5} when ± is minus. Subtract 5 from -1.

$$t=-\frac{6}{0.5}$$

Divide -6 by 0.5 by multiplying -6 by the reciprocal of 0.5.

$$t=-12$$

The equation is now solved.

$$t=8 t=-12$$

Steps for Completing the Square

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

$$t+0.25t^{2}=24$$

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.

$$0.25t^{2}+t=24$$

Multiply both sides by 4.

$$\frac{0.25t^{2}+t}{0.25}=\frac{24}{0.25}$$

Dividing by 0.25 undoes the multiplication by 0.25.

$$t^{2}+\frac{1}{0.25}t=\frac{24}{0.25}$$

Divide 1 by 0.25 by multiplying 1 by the reciprocal of 0.25.

$$t^{2}+4t=\frac{24}{0.25}$$

Divide 24 by 0.25 by multiplying 24 by the reciprocal of 0.25.

$$t^{2}+4t=96$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+4t+2^{2}=96+2^{2}$$

Square 2.

$$t^{2}+4t+4=96+4$$

Add 96 to 4.

$$t^{2}+4t+4=100$$

Factor t^{2}+4t+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+2\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+2\right)^{2}}=\sqrt{100}$$

Simplify.

$$t+2=10 t+2=-10$$

Subtract 2 from both sides of the equation.

$$t=8 t=-12$$