Solve 3d^2-51d+126

$$3 { d }^{ 2 } -51d+126$$

Factor

$$3\left(d-14\right)\left(d-3\right)$$

Solution Steps

Factor out 3.

$$3\left(d^{2}-17d+42\right)$$

Consider d^{2}-17d+42. Factor the expression by grouping. First, the expression needs to be rewritten as d^{2}+ad+bd+42. To find a and b, set up a system to be solved.

$$a+b=-17 ab=1\times 42=42$$

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 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -17.

$$a=-14 b=-3$$

Rewrite d^{2}-17d+42 as \left(d^{2}-14d\right)+\left(-3d+42\right).

$$\left(d^{2}-14d\right)+\left(-3d+42\right)$$

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

$$d\left(d-14\right)-3\left(d-14\right)$$

Factor out common term d-14 by using distributive property.

$$\left(d-14\right)\left(d-3\right)$$

Rewrite the complete factored expression.

$$3\left(d-14\right)\left(d-3\right)$$

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.

$$3d^{2}-51d+126=0$$

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.

$$d=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\times 3\times 126}}{2\times 3}$$

Square -51.

$$d=\frac{-\left(-51\right)±\sqrt{2601-4\times 3\times 126}}{2\times 3}$$

Multiply -4 times 3.

$$d=\frac{-\left(-51\right)±\sqrt{2601-12\times 126}}{2\times 3}$$

Multiply -12 times 126.

$$d=\frac{-\left(-51\right)±\sqrt{2601-1512}}{2\times 3}$$

Add 2601 to -1512.

$$d=\frac{-\left(-51\right)±\sqrt{1089}}{2\times 3}$$

Take the square root of 1089.

$$d=\frac{-\left(-51\right)±33}{2\times 3}$$

The opposite of -51 is 51.

$$d=\frac{51±33}{2\times 3}$$

Multiply 2 times 3.

$$d=\frac{51±33}{6}$$

Now solve the equation d=\frac{51±33}{6} when ± is plus. Add 51 to 33.

$$d=\frac{84}{6}$$

Divide 84 by 6.

$$d=14$$

Now solve the equation d=\frac{51±33}{6} when ± is minus. Subtract 33 from 51.

$$d=\frac{18}{6}$$

Divide 18 by 6.

$$d=3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 14 for x_{1} and 3 for x_{2}.

$$3d^{2}-51d+126=3\left(d-14\right)\left(d-3\right)$$

Evaluate

$$3\left(d-14\right)\left(d-3\right)$$