Solve 25x^2-5(5x)-84

$$25 { x }^{ 2 } -5(5x)-84$$

Factor

$$\left(5x-12\right)\left(5x+7\right)$$

Solution Steps

Multiply and combine like terms.

$$25x^{2}-25x-84$$

Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx-84. To find a and b, set up a system to be solved.

$$a+b=-25 ab=25\left(-84\right)=-2100$$

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 -2100.

$$1,-2100 2,-1050 3,-700 4,-525 5,-420 6,-350 7,-300 10,-210 12,-175 14,-150 15,-140 20,-105 21,-100 25,-84 28,-75 30,-70 35,-60 42,-50$$

Calculate the sum for each pair.

$$1-2100=-2099 2-1050=-1048 3-700=-697 4-525=-521 5-420=-415 6-350=-344 7-300=-293 10-210=-200 12-175=-163 14-150=-136 15-140=-125 20-105=-85 21-100=-79 25-84=-59 28-75=-47 30-70=-40 35-60=-25 42-50=-8$$

The solution is the pair that gives sum -25.

$$a=-60 b=35$$

Rewrite 25x^{2}-25x-84 as \left(25x^{2}-60x\right)+\left(35x-84\right).

$$\left(25x^{2}-60x\right)+\left(35x-84\right)$$

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

$$5x\left(5x-12\right)+7\left(5x-12\right)$$

Factor out common term 5x-12 by using distributive property.

$$\left(5x-12\right)\left(5x+7\right)$$

Evaluate

$$\left(5x-12\right)\left(5x+7\right)$$

Solution Steps

Multiply 5 and 5 to get 25.

$$25x^{2}-25x-84$$