Solve t^2-10t=11

$$t ^ { 2 } - 10 t = 11$$

Solve for t

$$t=-1$$

$$t=11$$

Steps Using Factoring

Subtract 11 from both sides.

$$t^{2}-10t-11=0$$

To solve the equation, factor t^{2}-10t-11 using formula t^{2}+\left(a+b\right)t+ab=\left(t+a\right)\left(t+b\right). To find a and b, set up a system to be solved.

$$a+b=-10 ab=-11$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-11 b=1$$

Rewrite factored expression \left(t+a\right)\left(t+b\right) using the obtained values.

$$\left(t-11\right)\left(t+1\right)$$

To find equation solutions, solve t-11=0 and t+1=0.

$$t=11 t=-1$$

Steps Using Factoring By Grouping

Subtract 11 from both sides.

$$t^{2}-10t-11=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as t^{2}+at+bt-11. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\left(-11\right)=-11$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-11 b=1$$

Rewrite t^{2}-10t-11 as \left(t^{2}-11t\right)+\left(t-11\right).

$$\left(t^{2}-11t\right)+\left(t-11\right)$$

Factor out t in t^{2}-11t.

$$t\left(t-11\right)+t-11$$

Factor out common term t-11 by using distributive property.

$$\left(t-11\right)\left(t+1\right)$$

To find equation solutions, solve t-11=0 and t+1=0.

$$t=11 t=-1$$

Steps Using the Quadratic Formula

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.

$$t^{2}-10t=11$$

Subtract 11 from both sides of the equation.

$$t^{2}-10t-11=11-11$$

Subtracting 11 from itself leaves 0.

$$t^{2}-10t-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-11\right)}}{2}$$

Square -10.

$$t=\frac{-\left(-10\right)±\sqrt{100-4\left(-11\right)}}{2}$$

Multiply -4 times -11.

$$t=\frac{-\left(-10\right)±\sqrt{100+44}}{2}$$

Add 100 to 44.

$$t=\frac{-\left(-10\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$t=\frac{-\left(-10\right)±12}{2}$$

The opposite of -10 is 10.

$$t=\frac{10±12}{2}$$

Now solve the equation t=\frac{10±12}{2} when ± is plus. Add 10 to 12.

$$t=\frac{22}{2}$$

Divide 22 by 2.

$$t=11$$

Now solve the equation t=\frac{10±12}{2} when ± is minus. Subtract 12 from 10.

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

Divide -2 by 2.

$$t=-1$$

The equation is now solved.

$$t=11 t=-1$$

Steps for Completing the Square

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.

$$t^{2}-10t=11$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-10t+\left(-5\right)^{2}=11+\left(-5\right)^{2}$$

Square -5.

$$t^{2}-10t+25=11+25$$

Add 11 to 25.

$$t^{2}-10t+25=36$$

Factor t^{2}-10t+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-5\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-5\right)^{2}}=\sqrt{36}$$

Simplify.

$$t-5=6 t-5=-6$$

Add 5 to both sides of the equation.

$$t=11 t=-1$$