Solve f(x)=15-2x-x^2

$$f ( x ) = 15 - 2 x - x ^ { 2 }$$

Factor

$$\left(3-x\right)\left(x+5\right)$$

Solution Steps

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

$$-x^{2}-2x+15$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-15=-15$$

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 -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=3 b=-5$$

Rewrite -x^{2}-2x+15 as \left(-x^{2}+3x\right)+\left(-5x+15\right).

$$\left(-x^{2}+3x\right)+\left(-5x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(-x+3\right)+5\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(x+5\right)$$

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.

$$-x^{2}-2x+15=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$x=\frac{-\left(-2\right)±\sqrt{4+60}}{2\left(-1\right)}$$

Add 4 to 60.

$$x=\frac{-\left(-2\right)±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-\left(-2\right)±8}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±8}{-2}$$

Now solve the equation x=\frac{2±8}{-2} when ± is plus. Add 2 to 8.

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

Now solve the equation x=\frac{2±8}{-2} when ± is minus. Subtract 8 from 2.

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

Divide -6 by -2.

$$x=3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and 3 for x_{2}.

$$-x^{2}-2x+15=-\left(x-\left(-5\right)\right)\left(x-3\right)$$

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

$$-x^{2}-2x+15=-\left(x+5\right)\left(x-3\right)$$

Evaluate

$$\left(3-x\right)\left(x+5\right)$$