Square Trinomial Calculator
Enter the coefficients (a, b, c) of your square trinomial in the form ax² + bx + c = 0 to find its roots, discriminant, vertex, and axis of symmetry.
0 = 0, which has infinite solutions.";
} else {
output = "This is the equation " + c + " = 0, which has no solution.";
}
} else {
var linearRoot = -c / b;
output = "Since 'a' is 0, this is a linear equation: " + b + "x + " + c + " = 0.";
output += "The single root is: x = " + linearRoot.toFixed(4) + "";
}
resultDiv.innerHTML = output;
return;
}
var discriminant = b * b – 4 * a * c;
output += "Discriminant (Δ): " + discriminant.toFixed(4) + "";
// Calculate roots
if (discriminant > 0) {
var root1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var root2 = (-b – Math.sqrt(discriminant)) / (2 * a);
output += "The trinomial has two distinct real roots:";
output += "x₁ = " + root1.toFixed(4) + "";
output += "x₂ = " + root2.toFixed(4) + "";
} else if (discriminant === 0) {
var root = -b / (2 * a);
output += "The trinomial has one real (repeated) root:";
output += "x = " + root.toFixed(4) + "";
} else { // discriminant < 0
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
output += "The trinomial has two complex conjugate roots:";
output += "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i";
output += "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
}
// Calculate vertex
var vertexX = -b / (2 * a);
var vertexY = a * vertexX * vertexX + b * vertexX + c;
output += "Vertex: (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")";
// Calculate axis of symmetry
output += "Axis of Symmetry: x = " + vertexX.toFixed(4) + "";
resultDiv.innerHTML = output;
}
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Understanding the Square Trinomial Calculator
A square trinomial is a polynomial of the second degree, meaning the highest power of the variable (usually 'x') is 2. It takes the general form: ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. These equations are fundamental in algebra and have wide applications in physics, engineering, economics, and many other fields.
What are the Coefficients?
- 'a' (Coefficient of x²): This determines the concavity of the parabola (the graph of a quadratic equation). If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. If 'a' is zero, the equation is no longer a quadratic but a linear equation.
- 'b' (Coefficient of x): This influences the position of the vertex and the axis of symmetry.
- 'c' (Constant Term): This represents the y-intercept of the parabola, i.e., where the graph crosses the y-axis (when x = 0).
The Quadratic Formula and Discriminant
The most common way to find the roots (or solutions) of a square trinomial is by using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
A crucial part of this formula is the expression under the square root, known as the discriminant (Δ):
Δ = b² - 4ac
The discriminant tells us about the nature of the roots without actually calculating them:
- 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 repeated 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 at all.
Vertex and Axis of Symmetry
The graph of a square trinomial is a parabola. The vertex is the highest or lowest point on the parabola. Its coordinates (xv, yv) can be found using the formulas:
- xv = -b / 2a
- yv = a(xv)² + b(xv) + c (substitute xv back into the original equation)
The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images. Its equation is simply:
x = -b / 2a
How to Use the Calculator
Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation ax² + bx + c = 0 into the respective fields. Click "Calculate" to instantly get the discriminant, the nature and values of the roots, the vertex coordinates, and the equation of the axis of symmetry.
Example Scenarios:
Example 1: Two Distinct Real Roots
Consider the equation: x² - 5x + 6 = 0
- a = 1
- b = -5
- c = 6
Calculation:
- Discriminant (Δ) = (-5)² – 4(1)(6) = 25 – 24 = 1
- Since Δ > 0, there are two real roots.
- x₁ = [5 + √1] / 2(1) = (5 + 1) / 2 = 3
- x₂ = [5 – √1] / 2(1) = (5 – 1) / 2 = 2
- Vertex x = -(-5) / 2(1) = 5/2 = 2.5
- Vertex y = (1)(2.5)² – 5(2.5) + 6 = 6.25 – 12.5 + 6 = -0.25
Results: Discriminant = 1, Roots: x₁ = 3, x₂ = 2, Vertex: (2.5, -0.25), Axis of Symmetry: x = 2.5
Example 2: One Real (Repeated) Root
Consider the equation: x² + 4x + 4 = 0
- a = 1
- b = 4
- c = 4
Calculation:
- Discriminant (Δ) = (4)² – 4(1)(4) = 16 – 16 = 0
- Since Δ = 0, there is one real repeated root.
- x = -4 / 2(1) = -2
- Vertex x = -(4) / 2(1) = -2
- Vertex y = (1)(-2)² + 4(-2) + 4 = 4 – 8 + 4 = 0
Results: Discriminant = 0, Root: x = -2, Vertex: (-2, 0), Axis of Symmetry: x = -2
Example 3: Complex Conjugate Roots
Consider the equation: x² + x + 1 = 0
- a = 1
- b = 1
- c = 1
Calculation:
- Discriminant (Δ) = (1)² – 4(1)(1) = 1 – 4 = -3
- Since Δ < 0, there are two complex conjugate roots.
- Real part = -1 / 2(1) = -0.5
- Imaginary part = √|-3| / 2(1) = √3 / 2 ≈ 0.866
- Vertex x = -(1) / 2(1) = -0.5
- Vertex y = (1)(-0.5)² + 1(-0.5) + 1 = 0.25 – 0.5 + 1 = 0.75
Results: Discriminant = -3, Roots: x₁ = -0.5 + 0.866i, x₂ = -0.5 – 0.866i, Vertex: (-0.5, 0.75), Axis of Symmetry: x = -0.5