Solve f(x)=-x^2-2x+24

$$f ( x ) = - x ^ { 2 } - 2 x + 24$$

Factor

$$\left(4-x\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-24=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=4 b=-6$$

Rewrite -x^{2}-2x+24 as \left(-x^{2}+4x\right)+\left(-6x+24\right).

$$\left(-x^{2}+4x\right)+\left(-6x+24\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x+4\right)+6\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(x+6\right)$$

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.

$$-x^{2}-2x+24=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-\left(-2\right)±\sqrt{4+96}}{2\left(-1\right)}$$

Add 4 to 96.

$$x=\frac{-\left(-2\right)±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-\left(-2\right)±10}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±10}{-2}$$

Now solve the equation x=\frac{2±10}{-2} when ± is plus. Add 2 to 10.

$$x=\frac{12}{-2}$$

Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{2±10}{-2} when ± is minus. Subtract 10 from 2.

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

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 -6 for x_{1} and 4 for x_{2}.

$$-x^{2}-2x+24=-\left(x-\left(-6\right)\right)\left(x-4\right)$$

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

$$-x^{2}-2x+24=-\left(x+6\right)\left(x-4\right)$$

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.

$$x ^ 2 +2x -24 = 0$$

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

$$r + s = -2 rs = -24$$

Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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>

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(-1 - u) (-1 + u) = -24$$

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

$$1 - u^2 = -24$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -24-1 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =-1 - 5 = -6 s = -1 + 5 = 4$$

Evaluate

$$\left(4-x\right)\left(x+6\right)$$