Solve (x-5)+(x^2-1)

$$( x - 5 ) + ( x ^ { 2 } - 1 )$$

Factor

$$\left(x-2\right)\left(x+3\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{2}+x-6$$

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-6\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 1.

$$a=-2 b=3$$

Rewrite x^{2}+x-6 as \left(x^{2}-2x\right)+\left(3x-6\right).

$$\left(x^{2}-2x\right)+\left(3x-6\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-2\right)+3\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+3\right)$$

Evaluate

$$\left(x-2\right)\left(x+3\right)$$

Solution Steps

Subtract 1 from -5 to get -6.

$$x-6+x^{2}$$