The cube of two consecutive numbers is 91.find the numbers.

Let the two consecutive numbers be \( x \) and \( x + 1 \).

The cube of two consecutive numbers is 91. Therefore, we can write the equation:

\[
x^3 + (x + 1)^3 = 91
\]

To solve for \( x \), we need to expand and simplify the equation. First, expand \( (x + 1)^3 \):

\[
(x + 1)^3 = x^3 + 3x^2 + 3x + 1
\]

Substitute this back into the original equation:

\[
x^3 + x^3 + 3x^2 + 3x + 1 = 91
\]

Combine like terms:

\[
2x^3 + 3x^2 + 3x + 1 = 91
\]

Subtract 91 from both sides to set the equation to zero:

\[
2x^3 + 3x^2 + 3x + 1 - 91 = 0
\]

Simplify the constant term:

\[
2x^3 + 3x^2 + 3x - 90 = 0
\]

Now, we need to find the integer value of \( x \) that satisfies this cubic equation. We can use trial and error for potential integer solutions or use more advanced methods like the Rational Root Theorem. 

Let's try \( x = 3 \):

\[
2(3)^3 + 3(3)^2 + 3(3) - 90 = 2(27) + 3(9) + 9 - 90
\]
\[
= 54 + 27 + 9 - 90
\]
\[
= 90 - 90
\]
\[
= 0
\]

So, \( x = 3 \) is a solution.

The two consecutive numbers are \( x = 3 \) and \( x + 1 = 4 \).

Therefore, the numbers are 3 and 4.

Do you want more details or have any questions?

Here are 8 related questions for further exploration:
1. What are the properties of cubic equations?
2. How do you expand binomials using the binomial theorem?
3. Can you solve the cubic equation using synthetic division?
4. What is the Rational Root Theorem and how is it applied?
5. How do you verify if a number is a solution to a polynomial equation?
6. What are the other methods to solve cubic equations?
7. How do consecutive numbers affect the solutions of polynomial equations?
8. What are some applications of solving polynomial equations in real life?

**Tip:** Always verify your solution by substituting it back into the original equation to ensure its correctness.

Learn how to find two consecutive numbers whose cubes sum up to 91 with a detailed step-by-step solution. This problem involves solving a cubic equation and applying algebraic concepts such as expansion and the Rational Root Theorem. Suitable for students in Grades 9-12.

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