Quadratic Equation Solver
Enter the coefficients for your quadratic equation in the form ax² + bx + c = 0 to find its roots, discriminant, and vertex. This is a fundamental concept often explored and visualized using tools like Desmos Graphing Calculator.
Calculation Results:
'; if (a === 0) { if (b === 0) { if (c === 0) { output += 'Equation: 0 = 0. This is true for all x (infinite solutions).'; } else { output += 'Equation: ' + c + ' = 0. This is a contradiction (no solutions).'; } } else { // Linear equation: bx + c = 0 var x = -c / b; output += 'This is a linear equation (a=0):' + b + 'x + ' + c + ' = 0';
output += 'Root (x): ' + x.toFixed(4) + '';
}
resultDiv.innerHTML = output;
return;
}
var discriminant = b * b – 4 * a * c;
output += 'Discriminant (Δ): ' + discriminant.toFixed(4) + '';
var x1, x2;
var rootType;
if (discriminant > 0) {
x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
rootType = 'Real and Distinct Roots';
output += 'Root 1 (x₁): ' + x1.toFixed(4) + '';
output += 'Root 2 (x₂): ' + x2.toFixed(4) + '';
} else if (discriminant === 0) {
x1 = -b / (2 * a);
rootType = 'Real and Equal Roots';
output += 'Root (x): ' + x1.toFixed(4) + ' (double root)';
} else { // discriminant < 0
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
rootType = 'Complex Conjugate Roots';
output += 'Root 1 (x₁): ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i';
output += 'Root 2 (x₂): ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i';
}
output += 'Root Type: ' + rootType + '';
// Calculate Vertex
var vertexX = -b / (2 * a);
var vertexY = a * vertexX * vertexX + b * vertexX + c;
output += 'Vertex: (' + vertexX.toFixed(4) + ', ' + vertexY.toFixed(4) + ')';
resultDiv.innerHTML = output;
}
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Understanding Quadratic Equations with Desmos
Quadratic equations are a cornerstone of algebra, appearing in various fields from physics to engineering and economics. They are polynomial equations of the second degree, meaning they contain at least one term where the variable is squared, but no term with a higher power. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients, and a cannot be zero.
The Significance of Coefficients (a, b, c)
- Coefficient 'a': This term dictates the parabola's direction and width. If
a > 0, the parabola opens upwards; ifa < 0, it opens downwards. A larger absolute value of 'a' results in a narrower parabola. - Coefficient 'b': The 'b' term influences the position of the parabola's vertex and axis of symmetry. It shifts the parabola horizontally.
- Coefficient 'c': This is the constant term, representing the y-intercept of the parabola. When
x = 0,y = c.
Solving Quadratic Equations: The Quadratic Formula
The most common method for finding the roots (or solutions) of a quadratic equation is the quadratic formula:
x = [-b ± sqrt(b² - 4ac)] / 2a
The term inside the square root, b² - 4ac, is called the discriminant (Δ). The value of the discriminant tells us about the nature of the roots:
- If
Δ > 0: There are two distinct real roots. The parabola intersects the x-axis at two different points. - If
Δ = 0: There is exactly one real root (a double root). The parabola touches the x-axis at exactly one point (its vertex). - If
Δ < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis.
The Vertex of a Parabola
The vertex is the highest or lowest point on the parabola, depending on whether it opens downwards or upwards. It represents the maximum or minimum value of the quadratic function. The coordinates of the vertex can be found using the formulas:
x-coordinate of vertex = -b / 2a
y-coordinate of vertex = f(-b / 2a) = a(-b/2a)² + b(-b/2a) + c
Visualizing with Desmos Graphing Calculator
Desmos is an incredibly powerful and intuitive online graphing calculator that makes visualizing quadratic equations (and many other functions) simple and engaging. Here's how Desmos enhances understanding:
- Instant Graphing: Simply type your quadratic equation (e.g.,
y = ax^2 + bx + c) into Desmos, and it instantly graphs the parabola. - Interactive Sliders: Desmos automatically creates sliders for coefficients
a,b, andc. You can drag these sliders to see in real-time how changing each coefficient affects the shape, position, and roots of the parabola. This dynamic interaction is invaluable for conceptual understanding. - Identifying Roots and Vertex: Desmos highlights key points on the graph, including the x-intercepts (roots), the y-intercept, and the vertex. Clicking on these points reveals their exact coordinates.
- Exploring Discriminant: While Desmos doesn't explicitly show the discriminant, by observing whether the parabola crosses the x-axis twice, once, or not at all, you can visually infer the nature of the roots.
Example Usage:
Consider the equation x² - 3x + 2 = 0. Using the calculator above:
- Coefficient a: 1
- Coefficient b: -3
- Coefficient c: 2
The calculator will show:
- Discriminant (Δ): 1
- Root 1 (x₁): 2.0000
- Root 2 (x₂): 1.0000
- Root Type: Real and Distinct Roots
- Vertex: (1.5000, -0.2500)
If you were to graph y = x² - 3x + 2 in Desmos, you would see the parabola crossing the x-axis at x=1 and x=2, and its lowest point (vertex) at (1.5, -0.25).
By combining a precise calculation tool like this solver with the visual power of Desmos, students and professionals alike can gain a deeper and more intuitive understanding of quadratic equations and their graphical representations.