Quadratic Formula Calculator Solve Quadratic Equation Quadratic Equation: ax² + b...
Quadratic Formula Calculator
Solve Quadratic Equation
Quadratic Equation: ax² + bx + c = 0
Coefficient a (x² term)
Coefficient b (x term)
Coefficient c (constant term)
💡 Quadratic Formula:
x = (-b ± √(b² - 4ac)) / (2a)
x = (-b ± √(b² - 4ac)) / (2a)
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Results
Equation:
x² - 5x + 6 = 0
Discriminant (D):
1
Nature of Roots:
Two Real and Distinct Roots
Root 1 (x₁):
3
Root 2 (x₂):
2
Sum of Roots:
5
Product of Roots:
6
Quadratic Function Graph
Vertex
x₁ = 3
x₂ = 2
Solution Steps:
1. Identify coefficients: a = 1, b = -5, c = 6
2. Calculate discriminant: D = b² - 4ac = 25 - 24 = 1
3. Apply quadratic formula: x = (5 ± √1) / 2
4. Solutions: x₁ = (5 + 1) / 2 = 3, x₂ = (5 - 1) / 2 = 2
Understanding Quadratic Equations
Quadratic equations are second-degree polynomial equations that have the general form ax² + bx + c = 0, where a ≠ 0. These equations are fundamental in algebra and appear frequently in physics, engineering, economics, and many other fields.
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides a systematic method to solve any quadratic equation. The term under the square root, b² - 4ac, is called the discriminant and determines the nature of the roots.
Three cases based on the discriminant:
- D > 0: Two real and distinct roots (the parabola intersects the x-axis at two points)
- D = 0: One real root (repeated root) - the parabola touches the x-axis at one point
- D < 0: Two complex conjugate roots - the parabola doesn't intersect the x-axis
This calculator helps students, teachers, and professionals quickly find solutions to quadratic equations and understand the underlying mathematical concepts.
Frequently Asked Questions
Q: What if 'a' is zero in my equation?
A: If the coefficient 'a' is zero, your equation is not quadratic but linear (bx + c = 0). The quadratic formula doesn't apply in this case. You should solve it as a linear equation: x = -c/b (provided b ≠ 0).
A: If the coefficient 'a' is zero, your equation is not quadratic but linear (bx + c = 0). The quadratic formula doesn't apply in this case. You should solve it as a linear equation: x = -c/b (provided b ≠ 0).
Q: Can I have decimal or fractional coefficients?
A: Yes, the quadratic formula works with any real numbers, including decimals and fractions. Simply enter your coefficients as decimal values (e.g., 0.5 for 1/2, 1.33 for 4/3). The calculator will handle the arithmetic correctly.
A: Yes, the quadratic formula works with any real numbers, including decimals and fractions. Simply enter your coefficients as decimal values (e.g., 0.5 for 1/2, 1.33 for 4/3). The calculator will handle the arithmetic correctly.
Q: What do complex roots look like?
A: When the discriminant is negative, you get complex roots in the form x = p ± qi, where i is the imaginary unit (√-1). For example, if D = -4, then √D = 2i, so the roots would be (-b ± 2i)/(2a). This calculator will display complex roots when they occur.
A: When the discriminant is negative, you get complex roots in the form x = p ± qi, where i is the imaginary unit (√-1). For example, if D = -4, then √D = 2i, so the roots would be (-b ± 2i)/(2a). This calculator will display complex roots when they occur.
Q: How can I verify my solutions are correct?
A: You can verify solutions by substituting them back into the original equation. If your solutions are correct, both should satisfy ax² + bx + c = 0. Additionally, you can use Vieta's formulas: the sum of roots should equal -b/a, and the product should equal c/a.
A: You can verify solutions by substituting them back into the original equation. If your solutions are correct, both should satisfy ax² + bx + c = 0. Additionally, you can use Vieta's formulas: the sum of roots should equal -b/a, and the product should equal c/a.
Q: What is the vertex of the parabola?
A: The vertex is the maximum or minimum point of the parabola, located at x = -b/(2a). The y-coordinate is found by substituting this x-value into the original equation. The vertex represents the turning point of the quadratic function.
A: The vertex is the maximum or minimum point of the parabola, located at x = -b/(2a). The y-coordinate is found by substituting this x-value into the original equation. The vertex represents the turning point of the quadratic function.
Q: When should I use factoring instead of the quadratic formula?
A: Factoring is faster when the quadratic factors easily with integer coefficients. However, the quadratic formula always works, even when factoring is difficult or impossible with rational numbers. Use the formula when you're unsure if factoring will work or when coefficients are decimals.
A: Factoring is faster when the quadratic factors easily with integer coefficients. However, the quadratic formula always works, even when factoring is difficult or impossible with rational numbers. Use the formula when you're unsure if factoring will work or when coefficients are decimals.
Q: What are some real-world applications of quadratic equations?
A: Quadratic equations model many real phenomena: projectile motion (height vs. time), profit maximization in business, area optimization problems, braking distance in physics, and parabolic reflectors in engineering. Understanding quadratics helps solve practical problems across many disciplines.
A: Quadratic equations model many real phenomena: projectile motion (height vs. time), profit maximization in business, area optimization problems, braking distance in physics, and parabolic reflectors in engineering. Understanding quadratics helps solve practical problems across many disciplines.