Solve f(x)=-3x^2-14x+24

$$f ( x ) = - 3 x ^ { 2 } - 14 x + 24$$

Factor

$$-\left(3x-4\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-14 ab=-3\times 24=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -14.

$$a=4 b=-18$$

Rewrite -3x^{2}-14x+24 as \left(-3x^{2}+4x\right)+\left(-18x+24\right).

$$\left(-3x^{2}+4x\right)+\left(-18x+24\right)$$

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

$$-x\left(3x-4\right)-6\left(3x-4\right)$$

Factor out common term 3x-4 by using distributive property.

$$\left(3x-4\right)\left(-x-6\right)$$

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.

$$-3x^{2}-14x+24=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-14\right)±\sqrt{196+12\times 24}}{2\left(-3\right)}$$

Multiply 12 times 24.

$$x=\frac{-\left(-14\right)±\sqrt{196+288}}{2\left(-3\right)}$$

Add 196 to 288.

$$x=\frac{-\left(-14\right)±\sqrt{484}}{2\left(-3\right)}$$

Take the square root of 484.

$$x=\frac{-\left(-14\right)±22}{2\left(-3\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±22}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{14±22}{-6}$$

Now solve the equation x=\frac{14±22}{-6} when ± is plus. Add 14 to 22.

$$x=\frac{36}{-6}$$

Divide 36 by -6.

$$x=-6$$

Now solve the equation x=\frac{14±22}{-6} when ± is minus. Subtract 22 from 14.

$$x=-\frac{8}{-6}$$

Reduce the fraction \frac{-8}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{3}$$

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 \frac{4}{3} for x_{2}.

$$-3x^{2}-14x+24=-3\left(x-\left(-6\right)\right)\left(x-\frac{4}{3}\right)$$

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

$$-3x^{2}-14x+24=-3\left(x+6\right)\left(x-\frac{4}{3}\right)$$

Subtract \frac{4}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-3x^{2}-14x+24=-3\left(x+6\right)\times \frac{-3x+4}{-3}$$

Cancel out 3, the greatest common factor in -3 and 3.

$$-3x^{2}-14x+24=\left(x+6\right)\left(-3x+4\right)$$

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.

$$x ^ 2 +\frac{14}{3}x -8 = 0$$

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

$$r + s = -\frac{14}{3} rs = -8$$

Two numbers r and s sum up to -\frac{14}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{14}{3} = -\frac{7}{3}. 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>

$$r = -\frac{7}{3} - u s = -\frac{7}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -8

$$(-\frac{7}{3} - u) (-\frac{7}{3} + u) = -8$$

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

$$\frac{49}{9} - u^2 = -8$$

Simplify the expression by subtracting \frac{49}{9} on both sides

$$-u^2 = -8-\frac{49}{9} = -\frac{121}{9}$$

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

$$u^2 = \frac{121}{9} u = \pm\sqrt{\frac{121}{9}} = \pm \frac{11}{3}$$

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

$$r =-\frac{7}{3} - \frac{11}{3} = -6 s = -\frac{7}{3} + \frac{11}{3} = 1.333$$

Evaluate

$$-\left(3x-4\right)\left(x+6\right)$$