Solve -33t+5t^2-56

$$- 33 t + 5 t ^ { 2 } - 56$$

Factor

$$\left(t-8\right)\left(5t+7\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$5t^{2}-33t-56$$

Factor the expression by grouping. First, the expression needs to be rewritten as 5t^{2}+at+bt-56. To find a and b, set up a system to be solved.

$$a+b=-33 ab=5\left(-56\right)=-280$$

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. List all such integer pairs that give product -280.

$$1,-280 2,-140 4,-70 5,-56 7,-40 8,-35 10,-28 14,-20$$

Calculate the sum for each pair.

$$1-280=-279 2-140=-138 4-70=-66 5-56=-51 7-40=-33 8-35=-27 10-28=-18 14-20=-6$$

The solution is the pair that gives sum -33.

$$a=-40 b=7$$

Rewrite 5t^{2}-33t-56 as \left(5t^{2}-40t\right)+\left(7t-56\right).

$$\left(5t^{2}-40t\right)+\left(7t-56\right)$$

Factor out 5t in the first and 7 in the second group.

$$5t\left(t-8\right)+7\left(t-8\right)$$

Factor out common term t-8 by using distributive property.

$$\left(t-8\right)\left(5t+7\right)$$

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.

$$5t^{2}-33t-56=0$$

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.

$$t=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 5\left(-56\right)}}{2\times 5}$$

Square -33.

$$t=\frac{-\left(-33\right)±\sqrt{1089-4\times 5\left(-56\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-\left(-33\right)±\sqrt{1089-20\left(-56\right)}}{2\times 5}$$

Multiply -20 times -56.

$$t=\frac{-\left(-33\right)±\sqrt{1089+1120}}{2\times 5}$$

Add 1089 to 1120.

$$t=\frac{-\left(-33\right)±\sqrt{2209}}{2\times 5}$$

Take the square root of 2209.

$$t=\frac{-\left(-33\right)±47}{2\times 5}$$

The opposite of -33 is 33.

$$t=\frac{33±47}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{33±47}{10}$$

Now solve the equation t=\frac{33±47}{10} when ± is plus. Add 33 to 47.

$$t=\frac{80}{10}$$

Divide 80 by 10.

$$t=8$$

Now solve the equation t=\frac{33±47}{10} when ± is minus. Subtract 47 from 33.

$$t=-\frac{14}{10}$$

Reduce the fraction \frac{-14}{10} to lowest terms by extracting and canceling out 2.

$$t=-\frac{7}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -\frac{7}{5} for x_{2}.

$$5t^{2}-33t-56=5\left(t-8\right)\left(t-\left(-\frac{7}{5}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$5t^{2}-33t-56=5\left(t-8\right)\left(t+\frac{7}{5}\right)$$

Add \frac{7}{5} to t by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5t^{2}-33t-56=5\left(t-8\right)\times \frac{5t+7}{5}$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5t^{2}-33t-56=\left(t-8\right)\left(5t+7\right)$$

Evaluate

$$\left(t-8\right)\left(5t+7\right)$$