Desmos Quadratic Equation Solver
Results will appear here.
Enter the coefficients for your quadratic equation (ax² + bx + c = 0) and click "Calculate" to find its roots, vertex, and other properties.Input Error
Please enter valid numbers for all coefficients.'; return; } if (a === 0) { resultDiv.style.backgroundColor = '#fff3cd'; resultDiv.style.borderColor = '#ffeeba'; resultDiv.style.color = '#856404'; if (b === 0) { if (c === 0) { output = 'Infinite Solutions
This is the equation 0 = 0, which is true for all x.'; } else { output = 'No Solution
This is the equation ' + c + ' = 0, which is false.'; } } else { var x_linear = -c / b; output = 'Linear Equation Detected (a=0)
'; output += 'This is a linear equation: ' + b + 'x + ' + c + ' = 0'; output += 'Solution (x): ' + x_linear.toFixed(4) + "; } resultDiv.innerHTML = output; return; } var discriminant = b * b – 4 * a * c; var x1, x2; var vertexX = -b / (2 * a); var vertexY = a * vertexX * vertexX + b * vertexX + c; output = 'Quadratic Properties
'; output += 'Equation: ' + a + 'x² + ' + b + 'x + ' + c + ' = 0'; output += 'Discriminant (Δ): ' + discriminant.toFixed(4) + "; if (discriminant >= 0) { x1 = (-b + Math.sqrt(discriminant)) / (2 * a); x2 = (-b – Math.sqrt(discriminant)) / (2 * a); output += 'Real Roots (x-intercepts):'; output += 'x₁ = ' + x1.toFixed(4) + "; output += 'x₂ = ' + x2.toFixed(4) + "; } else { var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); output += 'Complex Roots (No real x-intercepts):'; output += 'x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i'; output += 'x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } output += 'Vertex: (' + vertexX.toFixed(4) + ', ' + vertexY.toFixed(4) + ')'; output += 'Axis of Symmetry: x = ' + vertexX.toFixed(4) + "; var vertexFormSign = (vertexX >= 0) ? '-' : '+'; var vertexFormAbsX = Math.abs(vertexX); var vertexFormSignK = (vertexY >= 0) ? '+' : '-'; var vertexFormAbsY = Math.abs(vertexY); output += 'Vertex Form: y = ' + a + '(x ' + vertexFormSign + ' ' + vertexFormAbsX.toFixed(4) + ')² ' + vertexFormSignK + ' ' + vertexFormAbsY.toFixed(4) + "; resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = output; }Understanding Quadratic Equations with Desmos
Quadratic equations are fundamental in mathematics, describing parabolas that appear in everything from the trajectory of a projectile to the design of satellite dishes. A standard quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero.
What do the Coefficients Mean?
- Coefficient 'a': Determines the parabola's direction and width. If 'a' is positive, the parabola opens upwards; if negative, it opens downwards. A larger absolute value of 'a' makes the parabola narrower.
- Coefficient 'b': Influences the position of the parabola's vertex and axis of symmetry.
- Coefficient 'c': Represents the y-intercept of the parabola (where x=0).
Key Properties of a Parabola
When graphing a quadratic equation, several key properties help us understand its shape and position:
- Roots (x-intercepts): These are the points where the parabola crosses the x-axis (i.e., where y=0). A quadratic equation can have two real roots, one real root (a double root), or two complex roots (meaning it doesn't cross the x-axis).
- Discriminant (Δ): Calculated as
b² - 4ac, the discriminant tells us the nature of the roots:- If Δ > 0: Two distinct real roots.
- If Δ = 0: One real root (the parabola touches the x-axis at one point).
- If Δ < 0: Two complex roots (the parabola does not intersect the x-axis).
- Vertex: This is the highest or lowest point of the parabola. For a parabola opening upwards, it's the minimum point; for one opening downwards, it's the maximum point. The x-coordinate of the vertex is given by
-b / (2a). - Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is
x = -b / (2a). - Vertex Form: An alternative way to write a quadratic equation,
y = a(x - h)² + k, where (h, k) are the coordinates of the vertex. This form makes the vertex immediately apparent.
How This Calculator Helps with Desmos
Desmos is an incredibly powerful and intuitive online graphing calculator. While Desmos can graph any equation you input, understanding the underlying properties of a quadratic equation allows you to interpret your graphs more effectively and even predict their behavior.
This calculator provides you with all the critical parameters of a quadratic equation: the discriminant, roots, vertex coordinates, axis of symmetry, and the equation in vertex form. You can use these values to:
- Verify your graphs: Input your equation into Desmos and then check if the x-intercepts, vertex, and symmetry match the calculator's output.
- Explore transformations: See how changing 'a', 'b', or 'c' affects the calculated properties and then observe the corresponding changes in the Desmos graph.
- Quickly find key points: If you need to know the exact vertex or roots for a problem, this calculator provides them instantly, which you can then plot or label in Desmos.
- Understand vertex form: The calculator outputs the vertex form, which you can directly input into Desmos to see how it relates to the standard form graph.
Example Usage:
Let's use the default values in the calculator: a=1, b=-3, c=2.
This corresponds to the equation x² - 3x + 2 = 0.
Upon clicking "Calculate Quadratic Properties", you would get:
- Discriminant: 1 (since (-3)² – 4*1*2 = 9 – 8 = 1)
- Real Roots: x₁ = 2.0000, x₂ = 1.0000
- Vertex: (1.5000, -0.2500)
- Axis of Symmetry: x = 1.5000
- Vertex Form: y = 1(x – 1.5000)² – 0.2500
You can then input y = x^2 - 3x + 2 into Desmos and observe that the parabola indeed crosses the x-axis at 1 and 2, has its lowest point at (1.5, -0.25), and is symmetrical about the line x=1.5. You can also input y = (x - 1.5)^2 - 0.25 to see the identical graph.
Use this tool to deepen your understanding of quadratic functions and enhance your experience with graphing calculators like Desmos!