Question:
Let \( x=2 \) be a root of \( y = 4x^2 - 14x + q = 0 \). Then \( y \) is equal to:
Let \( x=2 \) be a root of \( y = 4x^2 - 14x + q = 0 \). Then \( y \) is equal to:
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Once you know \( q=12 \), you can quickly check the options by looking at the constant term. In Option (A), \( (-2) \times (-6) = +12 \). This matches our value of \( q \).
Updated On: May 1, 2026
- \( (x - 2)(4x - 6) \)
- \( (x - 2)(4x + 6) \)
- \( (x - 2)(-4x - 6) \)
- \( (x - 2)(-4x + 6) \)
- \( (x - 2)(4x + 3) \)
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The Correct Option is A
Solution and Explanation
Concept: If \( x = r \) is a root of a quadratic equation, then \( (x - r) \) is a factor of that equation. We can find the unknown constant \( q \) by substituting the root into the equation, and then factorize the resulting quadratic expression.
Step 1: Finding the value of \( q \).
Since \( x = 2 \) is a root, \( y(2) = 0 \): \[ 4(2)^2 - 14(2) + q = 0 \] \[ 4(4) - 28 + q = 0 \implies 16 - 28 + q = 0 \implies -12 + q = 0 \implies q = 12 \] The quadratic is \( y = 4x^2 - 14x + 12 \).
Step 2: Factorizing the expression.
We know one factor is \( (x - 2) \). We can use synthetic division or simply observe: \[ 4x^2 - 14x + 12 = 4x^2 - 8x - 6x + 12 \] \[ = 4x(x - 2) - 6(x - 2) = (x - 2)(4x - 6) \]
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