Quadratic Equations
Equal Roots
Grade 11
Question:
<p>If one root of equation \(x^2 + ax + 12 = 0\) is 4 while the equation \(x^2 + ax + b = 0\) has equal roots, then the value of b is</p>
<p>(a) 4</p>
<p>(b) \(\frac{49}{4}\)</p>
<p>(c) 7</p>
<p>(d) \(\frac{4}{7}\)</p>
Step-by-Step Solution
Key Concept: Use the fact that 4 is a root of the first equation to find 'a', then apply the discriminant condition (Δ = 0) for equal roots in the second equation to find 'b'.
<p><strong>Step 1:</strong> Since 4 is a root of x² + ax + 12 = 0, substitute x = 4:<br/>4² + a(4) + 12 = 0<br/>16 + 4a + 12 = 0<br/>28 + 4a = 0<br/>a = -7</p><p><strong>Step 2:</strong> Now we know a = -7. The second equation becomes:<br/>x² - 7x + b = 0</p><p><strong>Step 3:</strong> For the second equation to have equal roots, the discriminant must equal zero:<br/>Δ = b² - 4ac = 0<br/>(-7)² - 4(1)(b) = 0<br/>49 - 4b = 0<br/>4b = 49<br/>b = 49/4</p><p><strong>Verification:</strong> With b = 49/4, the equation x² - 7x + 49/4 = 0 has equal roots at x = 7/2 (using x = -b/2a for equal roots).</p><p><strong>∴ Answer:</strong> b</p>
Correct Answer: b