The equation 2x² + 3x - 5 = 0 has a = 2, b = 3, c = -5. What is the value of the discriminant (b² - 4ac)?
A-31
B9
C49
D-49
b² - 4ac = (3)² - 4(2)(-5) = 9 + 40 = 49. A common error is computing 4(2)(-5) as -40 instead of +40, since multiplying two negatives (the 4ac term with c = -5) yields a positive contribution when subtracted.
Question 2 True / False
When using the quadratic formula on x² - 6x + 9 = 0, you get x = (6 ± 0) / 2. This means the equation has two different real solutions.
TTrue
FFalse
Answer: False
When the discriminant equals zero, the ± contributes nothing — both 'solutions' collapse to the same value (x = 3). The quadratic has exactly one repeated root, not two distinct solutions. Geometrically, the parabola just touches the x-axis at one point.
Question 3 Short Answer
Why does the quadratic formula work on equations that factoring cannot easily solve?
Think about your answer, then reveal below.
Model answer: The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0, so it works for any values of a, b, and c — including irrational or complex roots that have no simple integer factors.
Factoring relies on finding integer (or simple rational) pairs that multiply to ac and add to b. Many quadratics have irrational roots (like sqrt(2)) that cannot be found by factoring. The quadratic formula bypasses this limitation because it is an algebraic derivation that handles all cases.