Questions: The sum of a number and 24 times its reciprocal is 11. Find the number(s).

The sum of a number and 24 times its reciprocal is 11. Find the number(s).
Transcript text: The sum of a number and 24 times its reciprocal is 11. Find the number(s).
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Solution

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Solution Steps

To solve the equation \( x^2 - 11x + 24 = 0 \), we need to factor the quadratic expression. We look for two numbers that multiply to 24 and add up to -11. Once we find these numbers, we can write the quadratic in its factored form and solve for \( x \).

Step 1: Set Up the Equation

We start with the quadratic equation derived from the problem statement: \[ x^2 - 11x + 24 = 0 \]

Step 2: Factor the Quadratic

Next, we factor the quadratic expression. We look for two numbers that multiply to \( 24 \) and add up to \( -11 \). The factors are \( -8 \) and \( -3 \): \[ (x - 8)(x - 3) = 0 \]

Step 3: Solve for \( x \)

Setting each factor equal to zero gives us the solutions: \[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

Final Answer

The solutions to the equation are: \[ \boxed{x = 3} \quad \text{and} \quad \boxed{x = 8} \]

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