--- title: "The Discriminant and the Number of Roots" description: "Use b² - 4ac to decide whether a quadratic has two roots, one repeated root or none, without solving it, and link that count to the graph crossing the x-axis." canonical: https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS source: https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Discriminant and the Number of Roots Use b² - 4ac to decide whether a quadratic has two roots, one repeated root or none, without solving it, and link that count to the graph crossing the x-axis. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS ## Ready when they can - Work out the discriminant and state how many real roots the equation has - Explain what a discriminant of zero means for the shape of the graph - Find the value of k that gives a quadratic equal roots ## Lesson: The Discriminant and the Number of Roots In the quadratic formula, the part under the root, b squared - 4ac, has a name: the discriminant. You have used the whole formula to solve equations. This stop reads that one part and asks what it tells you. The sign of the discriminant tells you how many real roots the equation has, before you solve anything. Positive gives two roots, zero gives one repeated root, and negative gives none. *(drawing: One calculation, three possible answers, no solving needed.)* Here is why the cases split. The formula splits the answer with plus or minus the square root of the discriminant. If that part is positive, the two values are genuinely different, so there are two roots. If it is zero, the plus and minus add nothing, and both paths land on the same number, -b divided by 2a. That single value is the repeated root. If the part is negative, the square root is not a real number, so the formula gives no real answers at all. **Example.** Watch the three cases happen. Start with x squared - 6x + 5. The discriminant is 36 - 20, which is 16, positive, so the curve crosses the x-axis twice. Now bump the constant up: x squared - 6x + 9. The discriminant is 36 - 36, which is zero, so the curve just touches the axis at its lowest point, one repeated root at x = 3. Bump it once more: x squared - 6x + 12. The discriminant is 36 - 48, which is -12, and the curve floats above the axis, never touching it. **Example.** Now put a letter in and ask when the roots repeat. When does x squared + kx + 9 = 0 have equal roots? Equal roots mean the discriminant is zero, so k squared - 36 = 0. That gives k squared = 36, so k = 6 or k = -6. Check with k = 6: the equation is x squared + 6x + 9, and its only root is x = -3. Both signs of k work because only k squared appears in the test. **Example.** The same test gives ranges. For x squared + kx + 9 = 0 to have no real roots, the discriminant must be negative: k squared - 36 < 0. That is k squared < 36, which happens when k sits between -6 and 6. The endpoints are the trap: at k = 6 or k = -6 the discriminant is zero, so the equation has one repeated root, not none. The inequality stays strict. **Tip.** If a question only asks for the number of roots, stop after the discriminant. There is no need to finish the formula. And when a question asks for equal roots, write 'discriminant = 0' before anything else. That one line sets the whole problem in motion. **Recap.** Work out b squared - 4ac: positive means two real roots, zero means one repeated root at -b over 2a, negative means none, and setting it to zero is how you find the k that makes the roots repeat. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY) ## Opens up - [Sketching Quadratic Graphs from Roots and the Turning Point](https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x) - [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv)