Standard problem: x² − 7x + 12 = 0
a=1, b=−7, c=12. Discriminant: 49 − 48 = 1. Roots: (7 ± 1)/2 = 4, 3. Verify: (x−3)(x−4) = x² − 7x + 12. ✓ Vertex: (3.5, −0.25) - the minimum point of the parabola.
Quadratic Equation Solver - 無料オンライン電卓
Solve any quadratic equation of the form ax² + bx + c = 0 - with real or complex roots. The solver also computes the discriminant, vertex, axis of symmetry, and identifies whether the parabola opens upward or downward. Useful for class 10 / class 12 board exam practice, entrance test prep, and physics projectile problems.
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x² − 5x + 6 = 0
Root 1
x₁ = 3
Root 2
x₂ = 2
Discriminant (Δ)
1
Δ > 0: two distinct real roots
Vertex
(2.5, -0.25)
Axis of symmetry
x = 2.5
Two distinct real roots. Parabola opens upward and crosses the x-axis at both roots.
一部のコンテンツはまだ英語で表示されます。
Quadratic Equation Solverとは
Quadratic equations are among the most frequently encountered equation types in school mathematics, physics, engineering and finance. CBSE Class 10 introduces them in depth (including by factoring, completing the square, and the quadratic formula); Class 11-12 extends the treatment to complex roots and parabolic geometry; JEE and other competitive exams test them every year. Beyond school, quadratics show up in projectile motion (kinematics), optimisation problems (where the vertex is the max or min), parabolic mirror and antenna design, and break-even analysis in business.
This solver handles the full case set - real distinct roots, repeated roots, and complex roots - and returns additional curve properties (vertex, axis of symmetry, parabola orientation) that are often part of the same problem. The quadratic formula x = (−b ± √(b² − 4ac))/2a works for every case, with the discriminant Δ = b² − 4ac determining the nature of roots: positive (two distinct real), zero (one repeated real), or negative (two complex conjugates).
Quadratic Equation Solverの使い方（5ステップ）
Enter coefficient a (the x² coefficient) - must be non-zero for it to be a quadratic.
Enter coefficient b (the x coefficient).
Enter coefficient c (the constant term).
Read both roots, the discriminant, and the parabola's vertex.
If the discriminant is negative, the calculator returns complex roots in p ± qi form.
Quadratic Equation Solverのしくみ
The quadratic formula always works: x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac decides the nature of roots. Positive → two distinct real roots. Zero → one repeated real root (the parabola just touches the x-axis). Negative → two complex roots that are conjugates of each other. The vertex coordinates are (−b/2a, c − b²/4a) and the axis of symmetry is x = −b/2a.
Formula
x = (−b ± √(b² − 4ac)) / 2a
- aCoefficient of x² (must be ≠ 0)
- bCoefficient of x
- cConstant term
- ΔDiscriminant b² − 4ac
- Δ > 0Two distinct real roots
- Δ = 0One repeated real root
- Δ < 0Two complex conjugate roots
Quadratic Equation Solverが結果を出す方法
The quadratic formula derives from completing the square on the general form ax² + bx + c = 0. Divide by a, move c/a to the right side, add (b/2a)² to both sides to complete the square, take the square root, solve for x. The result: x = (−b ± √(b² − 4ac))/2a. The ± gives the two roots; the discriminant under the square root tells you whether roots are real or complex.
Parabola geometry tied to quadratic coefficients: the parabola y = ax² + bx + c opens upward if a > 0 (vertex is the minimum) or downward if a < 0 (vertex is the maximum). Vertex x-coordinate = −b/(2a). Vertex y-coordinate = c − b²/(4a). Axis of symmetry: vertical line x = −b/(2a). Y-intercept: y = c (when x = 0). X-intercepts: the roots, if real.
Beyond the quadratic formula, factoring is often faster for nice integer-root quadratics. To factor ax² + bx + c when a = 1: find two numbers whose product is c and whose sum is b - those are the roots (with signs flipped). Example: x² + 5x + 6: find numbers multiplying to 6, summing to 5 - that's 2 and 3. So factors are (x + 2)(x + 3), roots are x = −2, −3. The calculator's quadratic-formula approach always works; factoring is a useful shortcut for clean cases.
計算例
Complex roots: x² + x + 1 = 0
a=1, b=1, c=1. Discriminant: 1 − 4 = −3 < 0. Roots: (−1 ± √3 i)/2 = −0.5 ± 0.866i. Two complex conjugate roots; the parabola never touches the x-axis.
Projectile motion: ball thrown upward at 20 m/s from 5m height - when hits ground?
Equation (taking g=10 m/s²): 5 + 20t − 5t² = 0, or −5t² + 20t + 5 = 0. Discriminant: 400 + 100 = 500. Roots: (−20 ± √500)/(−10) ≈ 4.24 sec (positive root). The ball hits ground after ~4.24 seconds.
Quadratic Equation Solverを使う場面
- Class 10 mathematics - quadratic equations chapter, board exam prep.
- Class 11-12 mathematics - extension to complex roots and parabolic geometry.
- JEE Main and JEE Advanced - quadratic and polynomial questions.
- NEET physics - projectile motion problems leading to quadratics.
- Engineering and physics - kinematics, optics (parabolic reflectors), antenna design.
- Business optimization - finding max profit or min cost points on quadratic profit functions.
- Computer graphics - bezier curves, collision detection involving quadratic paths.
よくある間違いと落とし穴
- Forgetting that a must be non-zero for a quadratic. If a = 0, the equation is linear, not quadratic.
- Sign errors in computing the discriminant. b² − 4ac, not b² + 4ac or b² − 4c.
- Missing the negative root. The ± gives two roots - both should be reported unless the problem context restricts (e.g., time can't be negative).
- Confusing vertex form with standard form. y = a(x−h)² + k has vertex (h, k); y = ax² + bx + c needs computation.
- Treating complex roots as 'no solution'. Complex roots are valid solutions - relevant in electrical engineering and AC circuit analysis.
このツールを使いこなす
- Always compute the discriminant first to know the nature of roots before applying the formula.
- For factoring practice, try integer factor pairs of c summing to b before going to the formula.
- For projectile and time problems, reject negative roots - they're mathematically valid but physically meaningless.
- Verify your roots by substituting back into the original equation. Both should give zero.
- Pair this calculator with the scientific, exponent and root calculators on this site for related algebra tasks.
Quadratic Equation Solverに関するよくある質問
What is a quadratic equation?
An equation of the form ax² + bx + c = 0 where a ≠ 0. "Quadratic" comes from the Latin quadratus (squared), referring to the x² term. Quadratic equations have exactly two roots in the complex number system, counted with multiplicity.
What is the discriminant?
The discriminant is b² − 4ac. Its sign tells you the nature of the roots without solving the equation: positive → two real roots, zero → one repeated real root, negative → two complex (non-real) roots.
How do I solve x² − 5x + 6 = 0?
a = 1, b = −5, c = 6. Discriminant = 25 − 24 = 1. Roots = (5 ± 1)/2 = 3 and 2. Verify: (x − 2)(x − 3) = x² − 5x + 6. ✓
What if the discriminant is negative?
The roots are complex conjugates. Example x² + x + 1 = 0: discriminant = 1 − 4 = −3. Roots = (−1 ± √3 i) / 2, which the calculator shows as −0.5 ± 0.866i.
What's the vertex of a parabola?
The peak (or valley) point of the curve y = ax² + bx + c. Its coordinates are x = −b/2a and y = c − b²/4a. If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, it opens downward and the vertex is the maximum.
What is the axis of symmetry?
A vertical line through the vertex. Its equation is x = −b/2a. The two roots (if real) are equidistant from this line.
Can I solve a linear equation here?
If a = 0 the equation is linear (bx + c = 0) and the calculator handles that case too - the answer is x = −c/b.
How do I factor a quadratic from its roots?
If r₁ and r₂ are the roots, the quadratic factors as a(x − r₁)(x − r₂). Example: roots 3 and 2 with a = 1 → (x − 3)(x − 2) = x² − 5x + 6.
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