Solve -4x^2-20x+96

$$- 4 x ^ { 2 } - 20 x + 96$$

Factor

$$4\left(3-x\right)\left(x+8\right)$$

Solution Steps

Factor out 4.

$$4\left(-x^{2}-5x+24\right)$$

Consider -x^{2}-5x+24. 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 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 -5.

$$a=3 b=-8$$

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

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

Factor out x in the first and 8 in the second group.

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

Factor out common term -x+3 by using distributive property.

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

Rewrite the complete factored expression.

$$4\left(-x+3\right)\left(x+8\right)$$

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.

$$-4x^{2}-20x+96=0$$

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.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-4\right)\times 96}}{2\left(-4\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-4\right)\times 96}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-20\right)±\sqrt{400+16\times 96}}{2\left(-4\right)}$$

Multiply 16 times 96.

$$x=\frac{-\left(-20\right)±\sqrt{400+1536}}{2\left(-4\right)}$$

Add 400 to 1536.

$$x=\frac{-\left(-20\right)±\sqrt{1936}}{2\left(-4\right)}$$

Take the square root of 1936.

$$x=\frac{-\left(-20\right)±44}{2\left(-4\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±44}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{20±44}{-8}$$

Now solve the equation x=\frac{20±44}{-8} when ± is plus. Add 20 to 44.

$$x=\frac{64}{-8}$$

Divide 64 by -8.

$$x=-8$$

Now solve the equation x=\frac{20±44}{-8} when ± is minus. Subtract 44 from 20.

$$x=-\frac{24}{-8}$$

Divide -24 by -8.

$$x=3$$

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

$$-4x^{2}-20x+96=-4\left(x-\left(-8\right)\right)\left(x-3\right)$$

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

$$-4x^{2}-20x+96=-4\left(x+8\right)\left(x-3\right)$$

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.

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

Evaluate

$$4\left(3-x\right)\left(x+8\right)$$