Solve 5-9x-2x^2

$$5 - 9 x - 2 x ^ { 2 }$$

Factor

$$-\left(2x-1\right)\left(x+5\right)$$

Solution Steps

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

$$-2x^{2}-9x+5$$

Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-2\times 5=-10$$

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 -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -9.

$$a=1 b=-10$$

Rewrite -2x^{2}-9x+5 as \left(-2x^{2}+x\right)+\left(-10x+5\right).

$$\left(-2x^{2}+x\right)+\left(-10x+5\right)$$

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

$$-x\left(2x-1\right)-5\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(-x-5\right)$$

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.

$$-2x^{2}-9x+5=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-9\right)±\sqrt{81+8\times 5}}{2\left(-2\right)}$$

Multiply 8 times 5.

$$x=\frac{-\left(-9\right)±\sqrt{81+40}}{2\left(-2\right)}$$

Add 81 to 40.

$$x=\frac{-\left(-9\right)±\sqrt{121}}{2\left(-2\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-9\right)±11}{2\left(-2\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±11}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{9±11}{-4}$$

Now solve the equation x=\frac{9±11}{-4} when ± is plus. Add 9 to 11.

$$x=\frac{20}{-4}$$

Divide 20 by -4.

$$x=-5$$

Now solve the equation x=\frac{9±11}{-4} when ± is minus. Subtract 11 from 9.

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

Reduce the fraction \frac{-2}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

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 \frac{1}{2} for x_{2}.

$$-2x^{2}-9x+5=-2\left(x-\left(-5\right)\right)\left(x-\frac{1}{2}\right)$$

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

$$-2x^{2}-9x+5=-2\left(x+5\right)\left(x-\frac{1}{2}\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-2x^{2}-9x+5=-2\left(x+5\right)\times \frac{-2x+1}{-2}$$

Cancel out 2, the greatest common factor in -2 and 2.

$$-2x^{2}-9x+5=\left(x+5\right)\left(-2x+1\right)$$

Evaluate

$$-\left(2x-1\right)\left(x+5\right)$$