Solve -x^2-9x-20

$$- x ^ { 2 } - 9 x - 20$$

Factor

$$-\left(x+4\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-\left(-20\right)=20$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -9.

$$a=-4 b=-5$$

Rewrite -x^{2}-9x-20 as \left(-x^{2}-4x\right)+\left(-5x-20\right).

$$\left(-x^{2}-4x\right)+\left(-5x-20\right)$$

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

$$x\left(-x-4\right)+5\left(-x-4\right)$$

Factor out common term -x-4 by using distributive property.

$$\left(-x-4\right)\left(x+5\right)$$

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.

$$-x^{2}-9x-20=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\left(-20\right)}}{2\left(-1\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)\left(-20\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4\left(-20\right)}}{2\left(-1\right)}$$

Multiply 4 times -20.

$$x=\frac{-\left(-9\right)±\sqrt{81-80}}{2\left(-1\right)}$$

Add 81 to -80.

$$x=\frac{-\left(-9\right)±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$x=\frac{-\left(-9\right)±1}{2\left(-1\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{9±1}{-2}$$

Now solve the equation x=\frac{9±1}{-2} when ± is plus. Add 9 to 1.

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

Divide 10 by -2.

$$x=-5$$

Now solve the equation x=\frac{9±1}{-2} when ± is minus. Subtract 1 from 9.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

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 -4 for x_{2}.

$$-x^{2}-9x-20=-\left(x-\left(-5\right)\right)\left(x-\left(-4\right)\right)$$

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

$$-x^{2}-9x-20=-\left(x+5\right)\left(x+4\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +9x +20 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -9 rs = 20$$

Two numbers r and s sum up to -9 exactly when the average of the two numbers is \frac{1}{2}*-9 = -\frac{9}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{9}{2} - u s = -\frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 20

$$(-\frac{9}{2} - u) (-\frac{9}{2} + u) = 20$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{81}{4} - u^2 = 20$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 20-\frac{81}{4} = -\frac{1}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{9}{2} - \frac{1}{2} = -5 s = -\frac{9}{2} + \frac{1}{2} = -4$$

Evaluate

$$-\left(x+4\right)\left(x+5\right)$$