Question

If the quadratic equation \(ax^2 + bx + c = 0\) has roots that differ by 4, and \(a + b + c = 12\), what is the value of \(b^2 - 4ac\)?

Hint:

The square of the difference of the roots of \(ax^2+bx+c=0\) is directly related to the discriminant (D) by \((\alpha - \beta)^2 = D/a^2\). This is a very useful shortcut.

Answer 1

Correct Answer: 16

Step 1: For \( ax^2+bx+c=0 \), the two roots differ by \( \frac{\sqrt{b^2-4ac}}{a} \).
Step 2: The roots differ by \( 4 \), so \( \frac{\sqrt{b^2-4ac}}{a}=4 \), giving \( b^2-4ac=16a^2 \).
Step 3: Taking the standard monic case \( a=1 \) (consistent with \( a+b+c=12 \) yielding valid integer roots such as \( 3,7 \) or \( -5,-1 \)):
\[ \boxed{b^2-4ac=16} \]