A quadratic equation is ax² + bx + c = 0. Solve using the quadratic formula: x = (-b ± √(b²-4ac)) / 2a. Example: x² - 5x + 6 = 0 (a=1, b=-5, c=6). Discriminant = 25-24 = 1. x = (5±1)/2. Solutions: x = 3 and x = 2. Check: (3)²-5(3)+6 = 9-15+6 = 0 ✓
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📐 Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 instantly. Get both roots, the discriminant, vertex, and complete step-by-step working. Perfect for students and teachers!
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ax² + bx + c = 0
💡 Example: for x²-5x+6=0, enter a=1, b=-5, c=6. Solutions are x=3 and x=2.
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The 365tool.net Quadratic Equation Solver uses the standard quadratic formula with full step-by-step working. Free for students, teachers, and engineers worldwide. No sign-up needed.
🤔 How Does This Work?
The Quadratic Equation Solver applies the quadratic formula:
x = (-b ± √(b²-4ac)) / 2a
Calculate discriminant D = b² - 4ac
If D > 0: two real roots x₁ = (-b+√D)/2a and x₂ = (-b-√D)/2a
If D = 0: one root x = -b/2a
If D < 0: two complex roots x = -b/2a ± i√|D|/2a
Vertex at x = -b/2a, y = c - b²/4a
❓ Frequently Asked Questions
What is a quadratic equation?▼
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The word 'quadratic' comes from the Latin 'quadratus' meaning square. These equations appear in physics (projectile motion), engineering, economics (profit optimization), and many real-world problems.
What is the discriminant?▼
The discriminant is b² - 4ac. If discriminant > 0: two distinct real roots. If discriminant = 0: one repeated real root (perfect square). If discriminant < 0: two complex (imaginary) roots. The discriminant tells you how many real solutions exist before you solve.
What are other ways to solve quadratic equations?▼
Factoring (when roots are integers): x²-5x+6 = (x-2)(x-3) = 0. Completing the square: add (b/2a)² to both sides. Quadratic formula: works for all cases. Graphing: find x-intercepts of y = ax²+bx+c. The quadratic formula always works.
What is the vertex of a parabola?▼
The vertex is the highest or lowest point of the parabola y = ax²+bx+c. Vertex x = -b/(2a). Vertex y = c - b²/(4a). If a>0, vertex is the minimum. If a<0, vertex is the maximum. The vertex form is y = a(x-h)²+k where (h,k) is the vertex.
Where do quadratic equations appear in real life?▼
Physics: calculating projectile motion, how long until a ball hits the ground. Engineering: bridge arch design, signal processing. Business: finding maximum profit or minimum cost. Architecture: parabolic arches and satellite dish designs. GPS: triangulating positions.
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The word 'quadratic' comes from the Latin 'quadratus' meaning square. These equations appear in physics (projectile motion), engineering, economics (profit optimization), and many real-world problems.
What is the discriminant?▼
The discriminant is b² - 4ac. If discriminant 0: two distinct real roots. If discriminant = 0: one repeated real root (perfect square). If discriminant 0: two complex (imaginary) roots. The discriminant tells you how many real solutions exist before you solve.
What are other ways to solve quadratic equations?▼
Factoring (when roots are integers): x²-5x+6 = (x-2)(x-3) = 0. Completing the square: add (b/2a)² to both sides. Quadratic formula: works for all cases. Graphing: find x-intercepts of y = ax²+bx+c. The quadratic formula always works.
What is the vertex of a parabola?▼
The vertex is the highest or lowest point of the parabola y = ax²+bx+c. Vertex x = -b/(2a). Vertex y = c - b²/(4a). If a0, vertex is the minimum. If a0, vertex is the maximum. The vertex form is y = a(x-h)²+k where (h,k) is the vertex.
Where do quadratic equations appear in real life?▼
Physics: calculating projectile motion, how long until a ball hits the ground. Engineering: bridge arch design, signal processing. Business: finding maximum profit or minimum cost. Architecture: parabolic arches and satellite dish designs. GPS: triangulating positions.