Discriminant and number of solutions — Digital SAT Math practice
Use b^2 - 4ac to count real solutions.
Key terms
- Discriminant: The value b^2 - 4ac that determines the quantity and type of roots in a quadratic equation.
- Real Solution: A valid x-value where the graph intersects or touches the horizontal axis.
- Double Root: A single real solution with multiplicity two, occurring when the discriminant equals zero.
- Complex Conjugate: Pair of non-real roots that appear together when the discriminant is negative.
- Parabola Vertex: The turning point of a quadratic graph located at x = -b / 2a on the axis of symmetry.
Common traps
- Forgetting to rewrite the equation in standard form before plugging coefficients into the discriminant formula.
- Confusing the discriminant value with the actual root values instead of using it only as a counting tool.
- Assuming D < 0 means zero solutions exist at all, rather than just no real solutions on the coordinate plane.
Key ideas to know
Calculating and Interpreting the Discriminant
For the quadratic equation 3x2 - 6x + 4 = 0, what is the discriminant D, and how many real solutions does it have?
D = (-6)2 - 4(3)(4) = 36 - 48 = -12. Since D < 0, there are no real solutions.
Watch out: Students often forget to square b or multiply all three coefficients correctly in 4ac. Always check signs carefully.
Finding Constants for Solution Counts
What value of k makes the equation x2 + kx + 9 = 0 have exactly one real solution?
Set D = 0. So k2 - 36 = 0 → k2 = 36 → k = ±6. Both values give a double root.
Watch out: Forgetting that k2 = 36 yields two possible values, not just one. Always solve completely for squared terms.
Misinterpreting D = 0 Geometrically
A student claims that if a parabola touches the x-axis at exactly one point, the discriminant must be negative. Is this correct? Explain why or why not.
The claim is wrong. Touching the x-axis at one point means D = 0, which gives exactly one real double root, not no real solutions.
Watch out: Confusing D < 0 (no real roots, graph never crosses axis) with D = 0 (one real root, graph just touches the axis).
Sample questions
Question 1
For what value of c does the equation x² + 6x + c = 0 have exactly one real solution?
- 3
- 6
- 9
- 36
Show answer and explanation
Answer: 9
Exactly one solution means the discriminant is 0: b² − 4ac = 36 − 4c = 0 → c = 9.