Solve 3d^2-26d+16

$$3 d ^ { 2 } - 26 d + 16$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3d^{2}+ad+bd+16. To find a and b, set up a system to be solved.

$$a+b=-26 ab=3\times 16=48$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both 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.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -26.

$$a=-24 b=-2$$

Rewrite 3d^{2}-26d+16 as \left(3d^{2}-24d\right)+\left(-2d+16\right).

$$\left(3d^{2}-24d\right)+\left(-2d+16\right)$$

Factor out 3d in the first and -2 in the second group.

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

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

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

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.

$$3d^{2}-26d+16=0$$

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.

$$d=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 3\times 16}}{2\times 3}$$

Square -26.

$$d=\frac{-\left(-26\right)±\sqrt{676-4\times 3\times 16}}{2\times 3}$$

Multiply -4 times 3.

$$d=\frac{-\left(-26\right)±\sqrt{676-12\times 16}}{2\times 3}$$

Multiply -12 times 16.

$$d=\frac{-\left(-26\right)±\sqrt{676-192}}{2\times 3}$$

Add 676 to -192.

$$d=\frac{-\left(-26\right)±\sqrt{484}}{2\times 3}$$

Take the square root of 484.

$$d=\frac{-\left(-26\right)±22}{2\times 3}$$

The opposite of -26 is 26.

$$d=\frac{26±22}{2\times 3}$$

Multiply 2 times 3.

$$d=\frac{26±22}{6}$$

Now solve the equation d=\frac{26±22}{6} when ± is plus. Add 26 to 22.

$$d=\frac{48}{6}$$

Divide 48 by 6.

$$d=8$$

Now solve the equation d=\frac{26±22}{6} when ± is minus. Subtract 22 from 26.

$$d=\frac{4}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$d=\frac{2}{3}$$

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

$$3d^{2}-26d+16=3\left(d-8\right)\left(d-\frac{2}{3}\right)$$

Subtract \frac{2}{3} from d by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3d^{2}-26d+16=3\left(d-8\right)\times \frac{3d-2}{3}$$

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

$$3d^{2}-26d+16=\left(d-8\right)\left(3d-2\right)$$

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 3

$$x ^ 2 -\frac{26}{3}x +\frac{16}{3} = 0$$

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

$$r + s = \frac{26}{3} rs = \frac{16}{3}$$

Two numbers r and s sum up to \frac{26}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{26}{3} = \frac{13}{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>

$$r = \frac{13}{3} - u s = \frac{13}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{16}{3}

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

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

$$\frac{169}{9} - u^2 = \frac{16}{3}$$

Simplify the expression by subtracting \frac{169}{9} on both sides

$$-u^2 = \frac{16}{3}-\frac{169}{9} = -\frac{121}{9}$$

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.

$$r =\frac{13}{3} - \frac{11}{3} = 0.667 s = \frac{13}{3} + \frac{11}{3} = 8$$

Evaluate

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