The discriminant
For x² + 2x + 5 = 0, b² − 4ac = 4 − 20 = −16. One negative number tells you, before solving anything, that the parabola never touches the x-axis.
Will the fountain jet reach 5 metres?
Picture a fountain in a park. The jet leaves a nozzle on the ground and lands 4 metres away, and at x metres from the nozzle its height is y = −x² + 4x metres. (The fountain and its numbers are made up.) Does the water reach 5 metres? Set −x² + 4x = 5 and tidy it up: x² − 4x + 5 = 0. Then b² − 4ac = 16 − 20 = −4. Negative. No real x works, so the water never gets that high. Try 3 metres instead. x² − 4x + 3 = 0 gives 16 − 12 = 4, so the jet passes that height twice, at x = 1 and x = 3. At 4 metres the number is 16 − 16 = 0, and the jet touches that height only once, at its top, x = 2. Not one root was needed to answer the first question, only the sign of b² − 4ac.
b² − 4ac decides how many solutions there are
In the quadratic formula, the expression b² − 4ac sits under the square root. It is called the discriminant, because it discriminates, or tells apart, the kinds of solutions an equation has. Its sign decides everything. If it is positive, the square root is a real number and the ± gives two different solutions. If it is zero, the ± adds and subtracts nothing, so both answers are the same number. If it is negative, there is no real square root at all, and the equation has no real solutions.
OpenStax refines the positive case into two. When b² − 4ac is a positive perfect square such as 1, 16 or 25, its square root is a whole number and the two solutions are rational, so the equation could also have been factored. When it is positive but not a perfect square, such as 5 or 17, the solutions contain an irrational square root. The table collects all four cases. In the negative case the solutions are complex numbers, which involve the square root of a negative number; they belong to a later course, and on a graph of real numbers they do not appear.
| b² − 4ac | Solutions | Example |
|---|---|---|
| positive square | two rational, 2 x-intercepts | x² − 5x + 6: 1 |
| positive, not square | two irrational, 2 x-intercepts | x² + x − 1: 5 |
| zero | one double root, touches at the vertex | x² − 4x + 4: 0 |
| negative | no real solutions, no x-intercepts | x² + 2x + 5: −16 |
Two crossings, one touch, or none
The solutions of ax² + bx + c = 0 are the x-intercepts of y = ax² + bx + c, so the discriminant also counts the x-intercepts. A positive discriminant means the parabola cuts through the x-axis twice. Zero means it just touches the axis at its vertex; that is why the vertex of y = (x − 3)² sits on the axis. A negative discriminant means the whole parabola stays above the axis (if a > 0) or below it (if a < 0). For y = 2x² − 3x + 5, b² − 4ac = 9 − 40 = −31, and since a > 0 the curve floats above the axis.
Working backwards: which c makes the curve touch the axis?
Because the discriminant is a formula in a, b and c, you can work backwards. Which c makes x² − 6x + c touch the x-axis exactly once? Set the discriminant to zero: 36 − 4c = 0, so c = 9, and indeed x² − 6x + 9 = (x − 3)². Which positive k gives x² + kx + 9 = 0 a double root? k² − 36 = 0, so k = 6. The fountain question used the same habit. Ask whether a solution exists before spending time on finding it.
The lab starts at y = x² − 4x + 4, whose discriminant is 0. Move c down by one and the discriminant becomes positive: two intercepts appear. Move c up and it turns negative: the parabola lifts off the axis. Watch the panel: it works out b² − 4ac with your numbers and names the case.
Drag to turn. Scroll or pinch to zoom.
- What is the discriminant of x² − 5x + 6 = 0?
- What is the discriminant of 3x² + 2x − 1 = 0?
- What is the discriminant of 2x² − 3x + 5 = 0?
- How many real solutions does x² − 4x + 4 = 0 have?
- One (a double root)
- Two
- None
- Infinitely many
- What kind of solutions does x² + x − 1 = 0 have?
- Two irrational
- Two rational
- One double root
- No real solutions
- What kind of solutions does x² + 2x + 5 = 0 have?
- No real solutions
- Two rational
- Two irrational
- One double root
- For which positive k does x² + kx + 9 = 0 have exactly one solution?
- How many x-intercepts does y = −x² + 4x − 5 have?
- Which c makes y = x² − 6x + c touch the x-axis exactly once?
- A quadratic equation has discriminant 49. What are its solutions like?
- Two rational solutions
- Two irrational solutions
- One double root
- No real solutions
- Writing b² = −25 when b = −5. Squaring a negative number gives a positive one: (−5)² = (−5) × (−5) = 25. For x² − 5x + 4 = 0 the slip gives −25 − 16 = −41 and the verdict "no real solutions", but the right value is 25 − 16 = 9, and 1 and 4 both work.
- Treating −4ac as negative when c is negative, as in x² + 4x − 5 = 0: 16 − 20 = −4. −4(1)(−5) = +20, so b² − 4ac = 16 + 20 = 36 and the solutions are 1 and −5. Whenever a and c have opposite signs, −4ac is positive, so the discriminant is positive and there are always two real solutions.
- Taking a discriminant of 0 to mean there is no solution. Zero means one solution, a double root. x² − 8x + 16 = 0 has b² − 4ac = 64 − 64 = 0 and x = 4. No real solution goes with a negative discriminant.
- Thinking a negative discriminant means negative solutions. The sign of the discriminant says nothing about the signs of the roots. x² + 4x + 3 = 0 has b² − 4ac = 16 − 12 = 4, which is positive, and both of its roots, −1 and −3, are negative.
- The discriminant is b² − 4ac, the part under the square root.
- Positive: two real solutions (rational if a perfect square). Zero: one double root. Negative: no real solutions.
- It also counts the x-intercepts: 2, 1 or 0.
- Set it to 0 to find the value that makes a parabola touch the axis.
Sources for this course (6)
- Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 중학교 1~3학년군 [9수02-19] 다항식의 곱셈과 인수분해(해설의 곱셈 공식 m(a + b), (a + b)², (a − b)², (a + b)(a − b), (x + a)(x + b), (ax + b)(cx + d)), [9수02-20] 이차방정식, [9수02-21] 이차함수의 개념, [9수02-22] 이차함수의 그래프(최댓값·최솟값은 x의 범위가 실수 전체인 경우만), 용어 인수·인수분해·완전제곱식·이차방정식·중근·근의 공식·이차함수·포물선·축·꼭짓점·최댓값·최솟값·평행이동, 「이차방정식은 해가 실수인 경우만」, 「근과 계수와의 관계는 다루지 않는다」, 「구한 해가 문제 상황에 적합한지 확인」; 고등학교 공통수학1 [10공수1-02-02] 실근과 허근·판별식, [10공수1-02-03] 근과 계수의 관계, [10공수1-02-06] 이차함수의 최대·최소와 실생활(제한된 범위, 공학 도구))
- Textbook: OpenStax, College Algebra 2e , 5.1 Quadratic Functions (parabola, vertex, axis of symmetry x = −b/(2a), general form ax² + bx + c, standard form a(x − h)² + k also called vertex form, f(x) = 2x² + 4x − 4 = 2(x + 1)² − 6, f(x) = −3(x + 2)² + 4, vertex of 2x² − 6x + 7 by −b/(2a) (the completing-the-square example x² − 6x + 7 in the course is our own), satellite dish cross-sections, 80 ft fence: maximum area 800 sq ft at L = 20, newspaper revenue maximum at $31.80 and $2,528,100, ball H(t) = −16t² + 80t + 40: top at 2.5 s, 140 ft, ground after about 5.458 s, rock H(t) = −16t² + 96t + 112: openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions)
- Textbook: OpenStax, Algebra and Trigonometry 2e, 2.5 Quadratic Equations (zero-product property, (x − 2)(x + 3) = x² + x − 6, square root property, completing the square, quadratic formula, discriminant table: openstax.org/books/algebra-and-trigonometry-2e/pages/2-5-quadratic-equations)
- Reference value: NIST CODATA standard acceleration of gravity 9.80665 m/s², about 32.174 ft/s² (physics.nist.gov/cgi-bin/cuu/Value?gn)
- Reference value: NIST, "Putting the Best Foot Forward" (1 foot = 0.3048 m exactly since 1959; nist.gov/pml/us-surveyfoot, checked 2026-10-19)
- History block only: MacTutor, University of St Andrews, Quadratic, cubic and quartic equations (Babylonians about 1800 BC, algorithmic completing the square with positive answers; Euclid about 300 BC geometric; Brahmagupta (598-665) admits negative quantities; al-Khwarizmi (c 800) six types, x² + 10x = 39; Abraham bar Hiyya, Liber embadorum 1145: mathshistory.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations).
Checked: 2026-10-18