Solve 10=10t+5t^2

$$10 = 10t+5 { t }^{ 2 }$$

Solve for t (complex solution)

$$t=\sqrt{3}-1\approx 0.732050808$$

$$t=-\left(\sqrt{3}+1\right)\approx -2.732050808$$

Steps Using the Quadratic Formula

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

$$10t+5t^{2}=10$$

Subtract 10 from both sides.

$$10t+5t^{2}-10=0$$

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.

$$5t^{2}+10t-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\times 5\left(-10\right)}}{2\times 5}$$

Square 10.

$$t=\frac{-10±\sqrt{100-4\times 5\left(-10\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-10±\sqrt{100-20\left(-10\right)}}{2\times 5}$$

Multiply -20 times -10.

$$t=\frac{-10±\sqrt{100+200}}{2\times 5}$$

Add 100 to 200.

$$t=\frac{-10±\sqrt{300}}{2\times 5}$$

Take the square root of 300.

$$t=\frac{-10±10\sqrt{3}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{-10±10\sqrt{3}}{10}$$

Now solve the equation t=\frac{-10±10\sqrt{3}}{10} when ± is plus. Add -10 to 10\sqrt{3}.

$$t=\frac{10\sqrt{3}-10}{10}$$

Divide -10+10\sqrt{3} by 10.

$$t=\sqrt{3}-1$$

Now solve the equation t=\frac{-10±10\sqrt{3}}{10} when ± is minus. Subtract 10\sqrt{3} from -10.

$$t=\frac{-10\sqrt{3}-10}{10}$$

Divide -10-10\sqrt{3} by 10.

$$t=-\sqrt{3}-1$$

The equation is now solved.

$$t=\sqrt{3}-1 t=-\sqrt{3}-1$$

Steps for Completing the Square

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

$$10t+5t^{2}=10$$

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.

$$5t^{2}+10t=10$$

Divide both sides by 5.

$$\frac{5t^{2}+10t}{5}=\frac{10}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\frac{10}{5}t=\frac{10}{5}$$

Divide 10 by 5.

$$t^{2}+2t=\frac{10}{5}$$

Divide 10 by 5.

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

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.

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

Square 1.

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

Add 2 to 1.

$$t^{2}+2t+1=3$$

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}.

$$\left(t+1\right)^{2}=3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{3}$$

Simplify.

$$t+1=\sqrt{3} t+1=-\sqrt{3}$$

Subtract 1 from both sides of the equation.

$$t=\sqrt{3}-1 t=-\sqrt{3}-1$$

Solve for t

$$t=\sqrt{3}-1\approx 0.732050808$$

$$t=-\sqrt{3}-1\approx -2.732050808$$

Steps Using the Quadratic Formula

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

$$10t+5t^{2}=10$$

Subtract 10 from both sides.

$$10t+5t^{2}-10=0$$

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.

$$5t^{2}+10t-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\times 5\left(-10\right)}}{2\times 5}$$

Square 10.

$$t=\frac{-10±\sqrt{100-4\times 5\left(-10\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-10±\sqrt{100-20\left(-10\right)}}{2\times 5}$$

Multiply -20 times -10.

$$t=\frac{-10±\sqrt{100+200}}{2\times 5}$$

Add 100 to 200.

$$t=\frac{-10±\sqrt{300}}{2\times 5}$$

Take the square root of 300.

$$t=\frac{-10±10\sqrt{3}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{-10±10\sqrt{3}}{10}$$

Now solve the equation t=\frac{-10±10\sqrt{3}}{10} when ± is plus. Add -10 to 10\sqrt{3}.

$$t=\frac{10\sqrt{3}-10}{10}$$

Divide -10+10\sqrt{3} by 10.

$$t=\sqrt{3}-1$$

Now solve the equation t=\frac{-10±10\sqrt{3}}{10} when ± is minus. Subtract 10\sqrt{3} from -10.

$$t=\frac{-10\sqrt{3}-10}{10}$$

Divide -10-10\sqrt{3} by 10.

$$t=-\sqrt{3}-1$$

The equation is now solved.

$$t=\sqrt{3}-1 t=-\sqrt{3}-1$$

Steps for Completing the Square

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

$$10t+5t^{2}=10$$

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.

$$5t^{2}+10t=10$$

Divide both sides by 5.

$$\frac{5t^{2}+10t}{5}=\frac{10}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\frac{10}{5}t=\frac{10}{5}$$

Divide 10 by 5.

$$t^{2}+2t=\frac{10}{5}$$

Divide 10 by 5.

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

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.

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

Square 1.

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

Add 2 to 1.

$$t^{2}+2t+1=3$$

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}.

$$\left(t+1\right)^{2}=3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{3}$$

Simplify.

$$t+1=\sqrt{3} t+1=-\sqrt{3}$$

Subtract 1 from both sides of the equation.

$$t=\sqrt{3}-1 t=-\sqrt{3}-1$$