Solve 9t^2-82t+185

$$9 { t }^{ 2 } -82t+185$$

Factor

$$\left(t-5\right)\left(9t-37\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9t^{2}+at+bt+185. To find a and b, set up a system to be solved.

$$a+b=-82 ab=9\times 185=1665$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 1665.

$$-1,-1665 -3,-555 -5,-333 -9,-185 -15,-111 -37,-45$$

Calculate the sum for each pair.

$$-1-1665=-1666 -3-555=-558 -5-333=-338 -9-185=-194 -15-111=-126 -37-45=-82$$

The solution is the pair that gives sum -82.

$$a=-45 b=-37$$

Rewrite 9t^{2}-82t+185 as \left(9t^{2}-45t\right)+\left(-37t+185\right).

$$\left(9t^{2}-45t\right)+\left(-37t+185\right)$$

Factor out 9t in the first and -37 in the second group.

$$9t\left(t-5\right)-37\left(t-5\right)$$

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

$$\left(t-5\right)\left(9t-37\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$9t^{2}-82t+185=0$$

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.

$$t=\frac{-\left(-82\right)±\sqrt{\left(-82\right)^{2}-4\times 9\times 185}}{2\times 9}$$

Square -82.

$$t=\frac{-\left(-82\right)±\sqrt{6724-4\times 9\times 185}}{2\times 9}$$

Multiply -4 times 9.

$$t=\frac{-\left(-82\right)±\sqrt{6724-36\times 185}}{2\times 9}$$

Multiply -36 times 185.

$$t=\frac{-\left(-82\right)±\sqrt{6724-6660}}{2\times 9}$$

Add 6724 to -6660.

$$t=\frac{-\left(-82\right)±\sqrt{64}}{2\times 9}$$

Take the square root of 64.

$$t=\frac{-\left(-82\right)±8}{2\times 9}$$

The opposite of -82 is 82.

$$t=\frac{82±8}{2\times 9}$$

Multiply 2 times 9.

$$t=\frac{82±8}{18}$$

Now solve the equation t=\frac{82±8}{18} when ± is plus. Add 82 to 8.

$$t=\frac{90}{18}$$

Divide 90 by 18.

$$t=5$$

Now solve the equation t=\frac{82±8}{18} when ± is minus. Subtract 8 from 82.

$$t=\frac{74}{18}$$

Reduce the fraction \frac{74}{18} to lowest terms by extracting and canceling out 2.

$$t=\frac{37}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and \frac{37}{9} for x_{2}.

$$9t^{2}-82t+185=9\left(t-5\right)\left(t-\frac{37}{9}\right)$$

Subtract \frac{37}{9} from t by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$9t^{2}-82t+185=9\left(t-5\right)\times \frac{9t-37}{9}$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9t^{2}-82t+185=\left(t-5\right)\left(9t-37\right)$$

Evaluate

$$\left(t-5\right)\left(9t-37\right)$$