Question

Solve the equation: d 3 − 12 d 2 − d + 12 = 0 Enter any solution as an integer or reduced fraction (not a mixed number or decimal), using a comma if there is more than one solution. If there is no solution, click No solution. Suggestion: Check any solution(s) in the original equation before submitting answer. No solution Hint: First factor by grouping. Then factor again if necessary.

Answer 1

Given the equation:

`d^3-12d^2-d+12=0`

To solve the equation for d, follow the steps below.

Step 01: Factor by grouping.

To factor by grouping, group the polynomial into two sections.

`\begin{gathered} d^3-d-12d^2+12=0 \\ \end{gathered}`

First group: d³ - d.

Second group: -12d² + 12.

The first group has "d" in common, while the second group has 12 (or -12) in common. To, factor out these values and write the expression again.

`d*\left(d^2-1\right)-12*\left(d^2-1\right)=0`

Factor out (d² - 1).

`(d^2-1)*(d-12)=0`

Step 02: Solve for d.

In order for the product to be zero, the first term or the second group must be zero.

`\begin{gathered} d^2-1=0 \\ or \\ d-12=0 \end{gathered}`

Solving the first equation:

`d^2-1=0`

Adding 1 to both sides and then taking the square root.

`\begin{gathered} d^2-1+1=0+1 \\ d^2=1 \\ √(d^2)=\pm√(1) \\ d=\pm1 \end{gathered}`

Solving the second equation:

`d-12=0`

Adding 12 to both sides:

`\begin{gathered} d-12+12=0+12 \\ d=12 \end{gathered}`

Answer: d = -1, 1, 12.