Solve 24d^2+39d-18

$$24 d ^ { 2 } + 39 d - 18$$

Factor

$$3\left(8d-3\right)\left(d+2\right)$$

Solution Steps

Factor out 3.

$$3\left(8d^{2}+13d-6\right)$$

Consider 8d^{2}+13d-6. Factor the expression by grouping. First, the expression needs to be rewritten as 8d^{2}+ad+bd-6. To find a and b, set up a system to be solved.

$$a+b=13 ab=8\left(-6\right)=-48$$

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 -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 13.

$$a=-3 b=16$$

Rewrite 8d^{2}+13d-6 as \left(8d^{2}-3d\right)+\left(16d-6\right).

$$\left(8d^{2}-3d\right)+\left(16d-6\right)$$

Factor out d in the first and 2 in the second group.

$$d\left(8d-3\right)+2\left(8d-3\right)$$

Factor out common term 8d-3 by using distributive property.

$$\left(8d-3\right)\left(d+2\right)$$

Rewrite the complete factored expression.

$$3\left(8d-3\right)\left(d+2\right)$$

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.

$$24d^{2}+39d-18=0$$

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.

$$d=\frac{-39±\sqrt{39^{2}-4\times 24\left(-18\right)}}{2\times 24}$$

Square 39.

$$d=\frac{-39±\sqrt{1521-4\times 24\left(-18\right)}}{2\times 24}$$

Multiply -4 times 24.

$$d=\frac{-39±\sqrt{1521-96\left(-18\right)}}{2\times 24}$$

Multiply -96 times -18.

$$d=\frac{-39±\sqrt{1521+1728}}{2\times 24}$$

Add 1521 to 1728.

$$d=\frac{-39±\sqrt{3249}}{2\times 24}$$

Take the square root of 3249.

$$d=\frac{-39±57}{2\times 24}$$

Multiply 2 times 24.

$$d=\frac{-39±57}{48}$$

Now solve the equation d=\frac{-39±57}{48} when ± is plus. Add -39 to 57.

$$d=\frac{18}{48}$$

Reduce the fraction \frac{18}{48} to lowest terms by extracting and canceling out 6.

$$d=\frac{3}{8}$$

Now solve the equation d=\frac{-39±57}{48} when ± is minus. Subtract 57 from -39.

$$d=-\frac{96}{48}$$

Divide -96 by 48.

$$d=-2$$

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

$$24d^{2}+39d-18=24\left(d-\frac{3}{8}\right)\left(d-\left(-2\right)\right)$$

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

$$24d^{2}+39d-18=24\left(d-\frac{3}{8}\right)\left(d+2\right)$$

Subtract \frac{3}{8} from d by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24d^{2}+39d-18=24\times \frac{8d-3}{8}\left(d+2\right)$$

Cancel out 8, the greatest common factor in 24 and 8.

$$24d^{2}+39d-18=3\left(8d-3\right)\left(d+2\right)$$

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.This is achieved by dividing both sides of the equation by 24

$$x ^ 2 +\frac{13}{8}x -\frac{3}{4} = 0$$

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

$$r + s = -\frac{13}{8} rs = -\frac{3}{4}$$

Two numbers r and s sum up to -\frac{13}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{8} = -\frac{13}{16}. 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>

$$r = -\frac{13}{16} - u s = -\frac{13}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{4}

$$(-\frac{13}{16} - u) (-\frac{13}{16} + u) = -\frac{3}{4}$$

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

$$\frac{169}{256} - u^2 = -\frac{3}{4}$$

Simplify the expression by subtracting \frac{169}{256} on both sides

$$-u^2 = -\frac{3}{4}-\frac{169}{256} = -\frac{361}{256}$$

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

$$u^2 = \frac{361}{256} u = \pm\sqrt{\frac{361}{256}} = \pm \frac{19}{16}$$

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

$$r =-\frac{13}{16} - \frac{19}{16} = -2 s = -\frac{13}{16} + \frac{19}{16} = 0.375$$

Evaluate

$$3\left(8d-3\right)\left(d+2\right)$$