Solve 25x^2-1+35x-7

$$25 { x }^{ 2 } -1+35x-7$$

Factor

$$\left(5x-1\right)\left(5x+8\right)$$

Solution Steps

Multiply and combine like terms.

$$25x^{2}+35x-8$$

Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=35 ab=25\left(-8\right)=-200$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -200.

$$-1,200 -2,100 -4,50 -5,40 -8,25 -10,20$$

Calculate the sum for each pair.

$$-1+200=199 -2+100=98 -4+50=46 -5+40=35 -8+25=17 -10+20=10$$

The solution is the pair that gives sum 35.

$$a=-5 b=40$$

Rewrite 25x^{2}+35x-8 as \left(25x^{2}-5x\right)+\left(40x-8\right).

$$\left(25x^{2}-5x\right)+\left(40x-8\right)$$

Factor out 5x in the first and 8 in the second group.

$$5x\left(5x-1\right)+8\left(5x-1\right)$$

Factor out common term 5x-1 by using distributive property.

$$\left(5x-1\right)\left(5x+8\right)$$

Evaluate

$$\left(5x-1\right)\left(5x+8\right)$$

Solution Steps

Subtract 7 from -1 to get -8.

$$25x^{2}-8+35x$$