Solve -t^2+10t-22=3

$$- t ^ { 2 } + 10 t - 22 = 3$$

Solve for t

$$t=5$$

Steps Using Factoring By Grouping

Subtract 3 from both sides.

$$-t^{2}+10t-22-3=0$$

Subtract 3 from -22 to get -25.

$$-t^{2}+10t-25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -t^{2}+at+bt-25. To find a and b, set up a system to be solved.

$$a+b=10 ab=-\left(-25\right)=25$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 25.

$$1,25 5,5$$

Calculate the sum for each pair.

$$1+25=26 5+5=10$$

The solution is the pair that gives sum 10.

$$a=5 b=5$$

Rewrite -t^{2}+10t-25 as \left(-t^{2}+5t\right)+\left(5t-25\right).

$$\left(-t^{2}+5t\right)+\left(5t-25\right)$$

Factor out -t in the first and 5 in the second group.

$$-t\left(t-5\right)+5\left(t-5\right)$$

Factor out common term t-5 by using distributive property.

$$\left(t-5\right)\left(-t+5\right)$$

To find equation solutions, solve t-5=0 and -t+5=0.

$$t=5 t=5$$

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.

$$-t^{2}+10t-22=3$$

Subtract 3 from both sides of the equation.

$$-t^{2}+10t-22-3=3-3$$

Subtracting 3 from itself leaves 0.

$$-t^{2}+10t-22-3=0$$

Subtract 3 from -22.

$$-t^{2}+10t-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 10 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Square 10.

$$t=\frac{-10±\sqrt{100-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-10±\sqrt{100+4\left(-25\right)}}{2\left(-1\right)}$$

Multiply 4 times -25.

$$t=\frac{-10±\sqrt{100-100}}{2\left(-1\right)}$$

Add 100 to -100.

$$t=\frac{-10±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$t=-\frac{10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=-\frac{10}{-2}$$

Divide -10 by -2.

$$t=5$$

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.

$$-t^{2}+10t-22=3$$

Add 22 to both sides of the equation.

$$-t^{2}+10t-22-\left(-22\right)=3-\left(-22\right)$$

Subtracting -22 from itself leaves 0.

$$-t^{2}+10t=3-\left(-22\right)$$

Subtract -22 from 3.

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

Divide both sides by -1.

$$\frac{-t^{2}+10t}{-1}=\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{10}{-1}t=\frac{25}{-1}$$

Divide 10 by -1.

$$t^{2}-10t=\frac{25}{-1}$$

Divide 25 by -1.

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

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.

$$t^{2}-10t+\left(-5\right)^{2}=-25+\left(-5\right)^{2}$$

Square -5.

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

Add -25 to 25.

$$t^{2}-10t+25=0$$

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}.

$$\left(t-5\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-5\right)^{2}}=\sqrt{0}$$

Simplify.

$$t-5=0 t-5=0$$

Add 5 to both sides of the equation.

$$t=5 t=5$$

The equation is now solved. Solutions are the same.

$$t=5$$