Solve s^2-6-s

$$s ^ { 2 } - 6 - s$$

Factor

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

Solution Steps

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

$$s^{2}-s-6$$

Factor the expression by grouping. First, the expression needs to be rewritten as s^{2}+as+bs-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 negative, the negative number has greater absolute value than the positive. 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.

$$a=-3 b=2$$

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

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

Factor out s in the first and 2 in the second group.

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

Factor out common term s-3 by using distributive property.

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

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.

$$s^{2}-s-6=0$$

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.

$$s=\frac{-\left(-1\right)±\sqrt{1-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$s=\frac{-\left(-1\right)±\sqrt{1+24}}{2}$$

Add 1 to 24.

$$s=\frac{-\left(-1\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$s=\frac{-\left(-1\right)±5}{2}$$

The opposite of -1 is 1.

$$s=\frac{1±5}{2}$$

Now solve the equation s=\frac{1±5}{2} when ± is plus. Add 1 to 5.

$$s=\frac{6}{2}$$

Divide 6 by 2.

$$s=3$$

Now solve the equation s=\frac{1±5}{2} when ± is minus. Subtract 5 from 1.

$$s=-\frac{4}{2}$$

Divide -4 by 2.

$$s=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -2 for x_{2}.

$$s^{2}-s-6=\left(s-3\right)\left(s-\left(-2\right)\right)$$

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

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

Evaluate

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