Solve f(x)=-x^2+5x+24

$$f ( x ) = - x ^ { 2 } + 5 x + 24$$

Factor

$$-\left(x-8\right)\left(x+3\right)$$

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.

$$a+b=5 ab=-24=-24$$

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 -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 5.

$$a=8 b=-3$$

Rewrite -x^{2}+5x+24 as \left(-x^{2}+8x\right)+\left(-3x+24\right).

$$\left(-x^{2}+8x\right)+\left(-3x+24\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-8\right)-3\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x-3\right)$$

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.

$$-x^{2}+5x+24=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-5±\sqrt{25+96}}{2\left(-1\right)}$$

Add 25 to 96.

$$x=\frac{-5±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-5±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±11}{-2}$$

Now solve the equation x=\frac{-5±11}{-2} when ± is plus. Add -5 to 11.

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

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{-5±11}{-2} when ± is minus. Subtract 11 from -5.

$$x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x=8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 8 for x_{2}.

$$-x^{2}+5x+24=-\left(x-\left(-3\right)\right)\left(x-8\right)$$

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

$$-x^{2}+5x+24=-\left(x+3\right)\left(x-8\right)$$

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.

$$x ^ 2 -5x -24 = 0$$

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

$$r + s = 5 rs = -24$$

Two numbers r and s sum up to 5 exactly when the average of the two numbers is \frac{1}{2}*5 = \frac{5}{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>

$$r = \frac{5}{2} - u s = \frac{5}{2} + u$$

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

$$(\frac{5}{2} - u) (\frac{5}{2} + u) = -24$$

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

$$\frac{25}{4} - u^2 = -24$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -24-\frac{25}{4} = -\frac{121}{4}$$

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

$$u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}$$

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

$$r =\frac{5}{2} - \frac{11}{2} = -3 s = \frac{5}{2} + \frac{11}{2} = 8$$

Evaluate

$$-\left(x-8\right)\left(x+3\right)$$