<p>The coefficient of x in the equation \(x^2 + px + q = 0\) was taken as 17 in place of 13 its roots were found to be -2 and -15. The roots of the original equation are</p>
Step-by-Step Solution
Key Concept: When the coefficient of x was incorrectly taken as 17 instead of 13, we can use Vieta's formulas on the incorrect equation to find q, then use the correct coefficient to find the actual roots.
<p><strong>Step 1:</strong> Write the equation with incorrect coefficient. When coefficient of x was taken as 17 instead of 13, the equation became: $x^2 + 17x + q = 0$ with roots -2 and -15.</p><p><strong>Step 2:</strong> Use Vieta's formulas on the incorrect equation. For $x^2 + 17x + q = 0$ with roots -2 and -15:
<ul><li>Sum of roots: $-2 + (-15) = -17$ ✓ (confirms coefficient is 17)</li><li>Product of roots: $(-2)(-15) = 30$</li></ul>
Therefore, $q = 30$.</p><p><strong>Step 3:</strong> Write the original equation with correct coefficient. The original equation is: $x^2 + 13x + 30 = 0$</p><p><strong>Step 4:</strong> Factor the original equation. We need two numbers that multiply to 30 and add to 13.
$$x^2 + 13x + 30 = (x + 5)(x + 8) = 0$$</p><p><strong>Step 5:</strong> Find the roots. From $(x + 5)(x + 8) = 0$:
$$x = -5 \text{ or } x = -8$$</p><p><strong>Verification:</strong> Sum = $-5 + (-8) = -13$ ✓ and Product = $(-5)(-8) = 30$ ✓</p><p><strong>∴ Answer:</strong> c</p>
Correct Answer: c