ACT Preparation / Unit 11 ACT 1008 worked example, 60 to 90 seconds
Using the Discriminant
The idea
Discriminant questions are answered in three steps. First, write the quadratic in standard form and identify a, b and c, which may contain an unknown constant. Second, write b² − 4ac. Third, set it greater than zero for two real solutions, equal to zero for exactly one, or less than zero for none, and solve for the constant.
In practice
Worked example: for what value of k does x² + kx + 9 = 0 have exactly one real solution? The discriminant is k² − 36. Setting it equal to zero gives k² = 36, so k = 6 or k = −6. Check: x² + 6x + 9 = (x + 3)², a perfect square with one repeated root.
Where people slip
Exactly one solution means the parabola touches the x-axis at its vertex, and the equation becomes a perfect square, which is a fast check.
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