Finding the Number: Sum of a Number, its Multiple of 5, and its Square

The problem asks us to find a specific number. We are given a condition involving this number: when the number is added to its multiple of 5 and its square, the total sum is 91.

Let's represent the unknown number using a variable. Let the number be $x$.

According to the problem description:

• The number itself is $x$.
• Its multiple of 5 is $5 \times x = 5x$.
• Its square is $x^2$.

The sum of these three quantities is given as 91. We can write this as an equation:

$\text{Number} + \text{Multiple of 5} + \text{Square} = 91$

$x + 5x + x^2 = 91$

Now, let's simplify and rearrange the equation to solve for $x$. Combine the terms involving $x$:

$6x + x^2 = 91$

To solve this, we can rearrange it into a standard quadratic equation form, which is $ax^2 + bx + c = 0$. Move the constant term (91) to the left side of the equation:

$x^2 + 6x - 91 = 0$

This is a quadratic equation. We can solve this equation by factoring, using the quadratic formula, or completing the square. Factoring seems suitable here.

We need to find two numbers that multiply to -91 and add up to 6. Let's list the factors of 91:

• 1 and 91
• 7 and 13

To get a product of -91 and a sum of +6, one of the factors must be negative. Let's try combinations of 7 and 13:

• $7 + (-13) = -6$ (Incorrect sum)
• $(-7) + 13 = 6$ (Correct sum)
• $(-7) \times 13 = -91$ (Correct product)

So, the two numbers are -7 and 13. We can use these to factor the quadratic equation:

$(x - 7)(x + 13) = 0$

For the product of two factors to be zero, at least one of the factors must be zero. So, we have two possible solutions:

• Case 1: $x - 7 = 0 \implies x = 7$
• Case 2: $x + 13 = 0 \implies x = -13$

The question asks for "the number". Let's check both solutions with the original condition.

Checking the Solutions

Check for $x = 7$:

• The number: 7
• Multiple of 5: $5 \times 7 = 35$
• Square: $7^2 = 49$
• Sum: $7 + 35 + 49 = 42 + 49 = 91$

This sum matches the condition (91). So, $x=7$ is a valid solution.

Check for $x = -13$:

• The number: -13
• Multiple of 5: $5 \times (-13) = -65$
• Square: $(-13)^2 = 169$
• Sum: $-13 + (-65) + 169 = -13 - 65 + 169 = -78 + 169 = 91$

This sum also matches the condition (91). So, $x=-13$ is also a valid mathematical solution to the equation.

However, looking at the options provided (9, 11, 7, 6), only 7 is listed. Therefore, 7 is the intended answer among the choices.

Verifying with the Given Options

Let's quickly check the other options just to be sure:

As verified, only the number 7 satisfies the given condition where the sum of the number, its multiple of 5, and its square is 91.

Thus, the number is 7.

Revision Table: Key Concepts

Additional Information: Solving Quadratic Equations

A quadratic equation in the form $ax^2 + bx + c = 0$ can be solved using various methods:

• Factoring: If the quadratic expression can be factored into $(px+q)(rx+s)$, the solutions are $x = -q/p$ and $x = -s/r$. This method is often the quickest when applicable, as in this problem.
• Quadratic Formula: The solutions are given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula works for any quadratic equation. For $x^2 + 6x - 91 = 0$, where $a=1$, $b=6$, $c=-91$:
  • $x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-91)}}{2(1)}$
  • $x = \frac{-6 \pm \sqrt{36 + 364}}{2}$
  • $x = \frac{-6 \pm \sqrt{400}}{2}$
  • $x = \frac{-6 \pm 20}{2}$
  • Two solutions: $x = \frac{-6 + 20}{2} = \frac{14}{2} = 7$ and $x = \frac{-6 - 20}{2} = \frac{-26}{2} = -13$.
• Completing the Square: This method involves transforming the equation so that one side is a perfect square trinomial. While always possible, it can be more involved than factoring or the quadratic formula.

In many word problems involving numbers, if multiple solutions arise from the algebra, context might imply a positive integer solution, especially if not specified otherwise. However, it's always best practice to check all mathematical solutions against the original problem statement.