Solve (a-a^2+6)=

$$( a - a ^ { 2 } + 6 ) =$$

Factor

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

Solution Steps

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

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

Factor the expression by grouping. First, the expression needs to be rewritten as -a^{2}+pa+qa+6. To find p and q, set up a system to be solved.

$$p+q=1 pq=-6=-6$$

Since pq is negative, p and q have the opposite signs. Since p+q 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.

$$p=3 q=-2$$

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

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

Factor out -a in the first and -2 in the second group.

$$-a\left(a-3\right)-2\left(a-3\right)$$

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

$$\left(a-3\right)\left(-a-2\right)$$

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.

$$-a^{2}+a+6=0$$

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.

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times 6.

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

Add 1 to 24.

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

Take the square root of 25.

$$a=\frac{-1±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-1±5}{-2}$$

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

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

Divide 4 by -2.

$$a=-2$$

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

$$a=-\frac{6}{-2}$$

Divide -6 by -2.

$$a=3$$

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

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

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

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

Evaluate

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