Solve 14+5x-x^2

$$14 + 5 x - x ^ { 2 }$$

Factor

$$-\left(x-7\right)\left(x+2\right)$$

Solution Steps

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

$$-x^{2}+5x+14$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=5 ab=-14=-14$$

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 -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 5.

$$a=7 b=-2$$

Rewrite -x^{2}+5x+14 as \left(-x^{2}+7x\right)+\left(-2x+14\right).

$$\left(-x^{2}+7x\right)+\left(-2x+14\right)$$

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

$$-x\left(x-7\right)-2\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x-2\right)$$

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.

$$-x^{2}+5x+14=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 14}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 14}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\times 14}}{2\left(-1\right)}$$

Multiply 4 times 14.

$$x=\frac{-5±\sqrt{25+56}}{2\left(-1\right)}$$

Add 25 to 56.

$$x=\frac{-5±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-5±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±9}{-2}$$

Now solve the equation x=\frac{-5±9}{-2} when ± is plus. Add -5 to 9.

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

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-5±9}{-2} when ± is minus. Subtract 9 from -5.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

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 7 for x_{2}.

$$-x^{2}+5x+14=-\left(x-\left(-2\right)\right)\left(x-7\right)$$

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

$$-x^{2}+5x+14=-\left(x+2\right)\left(x-7\right)$$

Evaluate

$$-\left(x-7\right)\left(x+2\right)$$