Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Find roots, discriminant, vertex, and view the parabola graph with step-by-step solution.
Roots
3, 2
Two real roots
Common quadratic equations
For the equation ax² + bx + c = 0
Your equation:
x² - 5x + 6 = 0
First Root (x₁)
3
Second Root (x₂)
2
Factored Form
(x -3)(x -2) = 0
Discriminant
1
b² - 4ac
Vertex
(2.5, -0.25)
Minimum point
Axis of Symmetry
x = 2.5
-b / (2a)
Y-Intercept
(0, 6)
Where x = 0
1. Identify coefficients:
a = 1, b = -5, c = 6
2. Calculate the discriminant (b² - 4ac):
Δ = (-5)² - 4(1)(6)
Δ = 25 - 24
Δ = 1
Since Δ > 0, there are two distinct real roots
3. Apply the quadratic formula:
x = (-b ± √Δ) / (2a)
x = (-(-5) ± √1) / (2 × 1)
x = (5 ± 1) / 2
4. Solutions:
x = 3, 2
Interactive formula breakdown
The values that satisfy the equation
Coefficient of x² term
Coefficient of x term
The constant value
b² - 4ac, determines root type
Test your understanding of quadratic equations
Practice with 6 problems to test your understanding.
Roots
3, 2
Two real roots
The quadratic formula x = (-b +/- sqrt(b^2-4ac)) / 2a solves any quadratic equation ax^2 + bx + c = 0. The discriminant (b^2-4ac) determines the nature of roots: if positive, two real roots exist; if zero, one repeated root; if negative, two complex conjugate roots. The vertex is at x = -b/(2a), and the parabola opens up when a > 0.
A quadratic equation is a polynomial equation of degree 2 in the form ax^2 + bx + c = 0, where a is not equal to 0. The quadratic formula provides a method to find the values of x (called roots or solutions) that make the equation true. These roots represent the x-intercepts of the corresponding parabola on a graph.
The quadratic formula x = (-b +/- sqrt(b^2-4ac)) / 2a solves any quadratic equation ax^2 + bx + c = 0. The discriminant (b^2-4ac) determines the nature of roots: if positive, two real roots exist; if zero, one repeated root; if negative, two complex conjugate roots. The vertex is at x = -b/(2a), and the parabola opens up when a > 0.
The quadratic formula x = (-b ± √(b²-4ac)) / 2a solves any quadratic equation ax² + bx + c = 0. It gives the x-values where the parabola crosses the x-axis (the roots or solutions).
The discriminant (b² - 4ac) determines the nature of the roots. If positive, there are two distinct real roots. If zero, there is one repeated real root. If negative, there are two complex conjugate roots.
The vertex is the turning point of the parabola - its highest or lowest point. For y = ax² + bx + c, the vertex is at x = -b/(2a), and the y-coordinate is found by substituting this x value.
The axis of symmetry is the vertical line that divides the parabola into two mirror images. It passes through the vertex and has the equation x = -b/(2a).
A parabola opens upward (∪ shape) when a > 0, with a minimum at the vertex. It opens downward (∩ shape) when a < 0, with a maximum at the vertex.
Complex roots occur when the discriminant is negative. They involve the imaginary unit i (where i² = -1) and always come in conjugate pairs like a + bi and a - bi. The parabola doesn't cross the x-axis.
Last updated: 2025-01-15
Roots
3, 2
Two real roots