Solve x+5x^2-25x-5

$$x+5 { x }^{ 2 } -25x-5$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$5x^{2}-24x-5$$

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

$$a+b=-24 ab=5\left(-5\right)=-25$$

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 -25.

$$1,-25 5,-5$$

Calculate the sum for each pair.

$$1-25=-24 5-5=0$$

The solution is the pair that gives sum -24.

$$a=-25 b=1$$

Rewrite 5x^{2}-24x-5 as \left(5x^{2}-25x\right)+\left(x-5\right).

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

Factor out 5x in 5x^{2}-25x.

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

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

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

Evaluate

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

Solution Steps

Combine x and -25x to get -24x.

$$-24x+5x^{2}-5$$