8.6The Discriminant and Nature of Roots
b² − 4ac — one calculation tells you whether a quadratic has two real solutions, one repeated solution, or no real solutions at all. No solving required.
Why This Matters
The discriminant is a shortcut that tells you everything about a quadratic's solutions before you solve — it's used in physics to determine if a projectile reaches a target, and on the SAT/ACT to quickly eliminate answer choices.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does the value of the discriminant b²−4ac determine the number and type of solutions of a quadratic equation, and how does this relate to the graph of the parabola?
Lesson Overview
The discriminant D = b²−4ac is the expression under the radical in the quadratic formula. Before you solve, compute D to predict the outcome: if D > 0 the parabola crosses the x-axis at two distinct points (two real roots); if D = 0 the parabola touches the x-axis at its vertex (one repeated real root); if D < 0 the parabola does not cross the x-axis at all (no real roots). This connection between algebra and geometry is one of the most powerful ideas in Algebra 1.
The Discriminant and the Parabola — Three Cases
D = b²−4ac > 0
Two distinct real roots
Parabola crosses x-axis at two distinct points.
D = b²−4ac = 0
One repeated real root
Parabola touches x-axis at vertex only.
D = b²−4ac < 0
No real roots
Parabola does not cross x-axis.
Worked Examples
Determine the nature of roots of x² − 5x + 6 = 0
a=1, b=−5, c=6
D = (−5)² − 4(1)(6) = 25 − 24 = 1
D = 1 > 0 → two distinct real roots
Solve: x = (5±1)/2 → x=3 or x=2
Determine the nature of roots of x² − 6x + 9 = 0
a=1, b=−6, c=9
D = (−6)² − 4(1)(9) = 36 − 36 = 0
D = 0 → one repeated real root
x = −b/(2a) = 6/2 = 3
Determine the nature of roots of x² + 2x + 5 = 0
a=1, b=2, c=5
D = (2)² − 4(1)(5) = 4 − 20 = −16
D = −16 < 0 → no real roots
Determine the nature of roots of 2x² − 3x + 1 = 0
a=2, b=−3, c=1
D = (−3)² − 4(2)(1) = 9 − 8 = 1
D = 1 > 0 → two distinct real roots
x = (3±1)/4 → x=1 or x=1/2
Find all values of k so that x² + kx + 9 = 0 has exactly one solution
For exactly one solution, D = 0
D = k² − 4(1)(9) = k² − 36
Set D = 0: k² − 36 = 0 → k² = 36
k = ±6
Guided Practice
Guided Practice Video: The Discriminant
Review how the discriminant determines the number and type of roots of a quadratic equation before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the discriminant of x² + 4x + 4 = 0 and state the nature of roots.
Hint: a=1, b=4, c=4. Compute D = 16 − 16. What is D equal to?
Find the discriminant of x² − 3x − 10 = 0 and state the nature of roots.
Hint: a=1, b=−3, c=−10. D = 9 + 40. Is D positive, zero, or negative?
Find the discriminant of 3x² + 2x + 4 = 0 and state the nature of roots.
Hint: a=3, b=2, c=4. D = 4 − 48. What is the sign of D?
For what value of c does x² − 4x + c = 0 have exactly one solution?
Hint: Set D = 0: (−4)² − 4(1)(c) = 0 → 16 − 4c = 0. Solve for c.
Without solving, determine the number of x-intercepts of y = x² + x + 1.
Hint: The x-intercepts correspond to solutions of x²+x+1=0. Compute D = 1 − 4.
Key Vocabulary
Discriminant
The expression D = b²−4ac in the quadratic formula. Its sign determines the number and type of solutions.
Example: For x²−5x+6=0: D = 25−24 = 1
Two Distinct Real Roots
When D > 0, the quadratic has two different real solutions and the parabola crosses the x-axis at two points.
Repeated Root (Double Root)
When D = 0, both solutions are equal: x = −b/(2a). The parabola touches the x-axis at its vertex.
Example: x²−6x+9=0 → x=3 (double root)
No Real Roots
When D < 0, there are no real solutions. The parabola lies entirely above or below the x-axis.
Nature of Roots
A description of the solutions of a quadratic: two distinct real, one repeated real, or no real roots.
Practice Questions
Interactive Practice — 5 Questions
What is the discriminant of a quadratic equation ax²+bx+c=0?
For x² − 5x + 6 = 0, what is the discriminant?
If D = 0, how many real solutions does the quadratic have?
For x² + 2x + 5 = 0, the discriminant is −16. What does this mean?
For x² + kx + 9 = 0 to have exactly one solution, k must equal:
Independent Practice
Independent Practice
Find the discriminant and state the nature of roots: x² − 7x + 10 = 0
Find the discriminant and state the nature of roots: x² + 4x + 4 = 0
Find the discriminant and state the nature of roots: 2x² + x + 3 = 0
Find the value of k so that x² − kx + 16 = 0 has exactly one solution.
Challenge: For what values of m does mx² − 6x + 3 = 0 have two distinct real roots? (Assume m ≠ 0.)
ChallengeCommon Mistakes
Computing D = b² − 4ac but forgetting to square b first — e.g., writing D = b − 4ac.
Always square b: D = b² − 4ac. If b = −5, then b² = 25, not −25.
Confusing D = 0 with 'no solution' — writing 'no solution' when D = 0.
D = 0 means exactly one real solution (a double root), not zero solutions.
Using the wrong sign for c — e.g., for x² + 3x − 4 = 0 writing c = +4 instead of c = −4.
Read c directly from standard form ax²+bx+c=0. Here c = −4, so D = 9 − 4(1)(−4) = 9 + 16 = 25.
Forgetting that 4ac means 4 × a × c — computing D = b² − 4 + a + c instead.
D = b² − (4)(a)(c). Multiply all three: 4, a, and c together before subtracting.
Math Tips
Always compute the discriminant before using the full quadratic formula — it tells you how many solutions to expect and can save time.
SAT/ACT tip: if a question asks for the number of x-intercepts of a parabola, just compute D = b²−4ac. No solving needed.
D > 0 and a perfect square → rational roots (nice fractions or integers). D > 0 but not a perfect square → irrational roots.
The discriminant also tells you about the vertex: D = 0 means the vertex is on the x-axis; D < 0 means the vertex is above (opens up) or below (opens down) the x-axis.