Question:
Given the equations:
\(x^2 + 5x - 24 = 0\)
\(y^2 - 9y + 20 = 0\)
Quantity A: x
Quantity B: y
Given the equations:
\(x^2 + 5x - 24 = 0\)
\(y^2 - 9y + 20 = 0\)
Quantity A: x
Quantity B: y
\(x^2 + 5x - 24 = 0\)
\(y^2 - 9y + 20 = 0\)
Quantity A: x
Quantity B: y
Show Hint
In quantitative comparison questions involving variables with multiple possible values, check the extreme values. Compare the maximum possible value of one quantity with the minimum possible value of the other. If a consistent relationship holds (e.g., max(A)<min(B)), you have your answer. If not, the relationship cannot be determined.
Updated On: Oct 3, 2025
- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
We are asked to compare the possible values of x and y, which are the solutions (roots) of two separate quadratic equations. We must solve each equation to find the possible values for x and y.
Step 2: Key Formula or Approach:
We will solve each quadratic equation by factoring. For an equation of the form \(ax^2 + bx + c = 0\), we look for two numbers that multiply to \(c\) and add to \(b\).
Step 3: Detailed Explanation:
Solving for x:
The equation for x is \(x^2 + 5x - 24 = 0\).
We need two numbers that multiply to -24 and add to +5. These numbers are +8 and -3.
So, we can factor the equation as: \[ (x + 8)(x - 3) = 0 \] This gives two possible values for x: \[ x = -8 \quad \text{or} \quad x = 3 \]
Solving for y:
The equation for y is \(y^2 - 9y + 20 = 0\).
We need two numbers that multiply to +20 and add to -9. These numbers are -4 and -5.
So, we can factor the equation as: \[ (y - 4)(y - 5) = 0 \] This gives two possible values for y: \[ y = 4 \quad \text{or} \quad y = 5 \]
Comparing Quantity A and Quantity B:
The possible values for Quantity A (x) are \{-8, 3\}.
The possible values for Quantity B (y) are \{4, 5\}.
Let's compare the largest possible value of x with the smallest possible value of y.
The largest value for x is 3.
The smallest value for y is 4.
Since \(3<4\), any possible value of y will be greater than any possible value of x.
Step 4: Final Answer:
Quantity B is always greater than Quantity A.
We are asked to compare the possible values of x and y, which are the solutions (roots) of two separate quadratic equations. We must solve each equation to find the possible values for x and y.
Step 2: Key Formula or Approach:
We will solve each quadratic equation by factoring. For an equation of the form \(ax^2 + bx + c = 0\), we look for two numbers that multiply to \(c\) and add to \(b\).
Step 3: Detailed Explanation:
Solving for x:
The equation for x is \(x^2 + 5x - 24 = 0\).
We need two numbers that multiply to -24 and add to +5. These numbers are +8 and -3.
So, we can factor the equation as: \[ (x + 8)(x - 3) = 0 \] This gives two possible values for x: \[ x = -8 \quad \text{or} \quad x = 3 \]
Solving for y:
The equation for y is \(y^2 - 9y + 20 = 0\).
We need two numbers that multiply to +20 and add to -9. These numbers are -4 and -5.
So, we can factor the equation as: \[ (y - 4)(y - 5) = 0 \] This gives two possible values for y: \[ y = 4 \quad \text{or} \quad y = 5 \]
Comparing Quantity A and Quantity B:
The possible values for Quantity A (x) are \{-8, 3\}.
The possible values for Quantity B (y) are \{4, 5\}.
Let's compare the largest possible value of x with the smallest possible value of y.
The largest value for x is 3.
The smallest value for y is 4.
Since \(3<4\), any possible value of y will be greater than any possible value of x.
Step 4: Final Answer:
Quantity B is always greater than Quantity A.
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