Solve 2t=-5+t^2

$$2 t = - 5 + t ^ { 2 }$$

Solve for t

$$t=\sqrt{6}+1\approx 3.449489743$$

$$t=1-\sqrt{6}\approx -1.449489743$$

Steps Using the Quadratic Formula

Subtract -5 from both sides.

$$2t-\left(-5\right)=t^{2}$$

The opposite of -5 is 5.

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

Subtract t^{2} from both sides.

$$2t+5-t^{2}=0$$

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.

$$-t^{2}+2t+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Square 2.

$$t=\frac{-2±\sqrt{4-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-2±\sqrt{4+4\times 5}}{2\left(-1\right)}$$

Multiply 4 times 5.

$$t=\frac{-2±\sqrt{4+20}}{2\left(-1\right)}$$

Add 4 to 20.

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

Take the square root of 24.

$$t=\frac{-2±2\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-2±2\sqrt{6}}{-2}$$

Now solve the equation t=\frac{-2±2\sqrt{6}}{-2} when ± is plus. Add -2 to 2\sqrt{6}.

$$t=\frac{2\sqrt{6}-2}{-2}$$

Divide -2+2\sqrt{6} by -2.

$$t=1-\sqrt{6}$$

Now solve the equation t=\frac{-2±2\sqrt{6}}{-2} when ± is minus. Subtract 2\sqrt{6} from -2.

$$t=\frac{-2\sqrt{6}-2}{-2}$$

Divide -2-2\sqrt{6} by -2.

$$t=\sqrt{6}+1$$

The equation is now solved.

$$t=1-\sqrt{6} t=\sqrt{6}+1$$

Steps for Completing the Square

Subtract t^{2} from both sides.

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

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.

$$-t^{2}+2t=-5$$

Divide both sides by -1.

$$\frac{-t^{2}+2t}{-1}=-\frac{5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{2}{-1}t=-\frac{5}{-1}$$

Divide 2 by -1.

$$t^{2}-2t=-\frac{5}{-1}$$

Divide -5 by -1.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-2t+1=5+1$$

Add 5 to 1.

$$t^{2}-2t+1=6$$

Factor t^{2}-2t+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-1\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-1\right)^{2}}=\sqrt{6}$$

Simplify.

$$t-1=\sqrt{6} t-1=-\sqrt{6}$$

Add 1 to both sides of the equation.

$$t=\sqrt{6}+1 t=1-\sqrt{6}$$