Solve 72+b-b^2

$$72 + b - b ^ { 2 }$$

Factor

$$-\left(b-9\right)\left(b+8\right)$$

Solution Steps

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

$$-b^{2}+b+72$$

Factor the expression by grouping. First, the expression needs to be rewritten as -b^{2}+pb+qb+72. To find p and q, set up a system to be solved.

$$p+q=1 pq=-72=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 1.

$$p=9 q=-8$$

Rewrite -b^{2}+b+72 as \left(-b^{2}+9b\right)+\left(-8b+72\right).

$$\left(-b^{2}+9b\right)+\left(-8b+72\right)$$

Factor out -b in the first and -8 in the second group.

$$-b\left(b-9\right)-8\left(b-9\right)$$

Factor out common term b-9 by using distributive property.

$$\left(b-9\right)\left(-b-8\right)$$

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.

$$-b^{2}+b+72=0$$

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.

$$b=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 72}}{2\left(-1\right)}$$

Square 1.

$$b=\frac{-1±\sqrt{1-4\left(-1\right)\times 72}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-1±\sqrt{1+4\times 72}}{2\left(-1\right)}$$

Multiply 4 times 72.

$$b=\frac{-1±\sqrt{1+288}}{2\left(-1\right)}$$

Add 1 to 288.

$$b=\frac{-1±\sqrt{289}}{2\left(-1\right)}$$

Take the square root of 289.

$$b=\frac{-1±17}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-1±17}{-2}$$

Now solve the equation b=\frac{-1±17}{-2} when ± is plus. Add -1 to 17.

$$b=\frac{16}{-2}$$

Divide 16 by -2.

$$b=-8$$

Now solve the equation b=\frac{-1±17}{-2} when ± is minus. Subtract 17 from -1.

$$b=-\frac{18}{-2}$$

Divide -18 by -2.

$$b=9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -8 for x_{1} and 9 for x_{2}.

$$-b^{2}+b+72=-\left(b-\left(-8\right)\right)\left(b-9\right)$$

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

$$-b^{2}+b+72=-\left(b+8\right)\left(b-9\right)$$

Evaluate

$$-\left(b-9\right)\left(b+8\right)$$