Quadratic Equation Solver
Enter the coefficients a, b, and c for your quadratic equation in the standard form ax² + bx + c = 0 to find its roots and a step-by-step solution.
Solution Steps:
"; // Step 1: Identify coefficients solutionSteps += "Step 1: Identify the coefficients."; solutionSteps += "Given the quadratic equation in the formax² + bx + c = 0:";
solutionSteps += "a = " + a + "";
solutionSteps += "b = " + b + "";
solutionSteps += "c = " + c + "";
// Handle case where 'a' is 0 (linear equation or simpler)
if (a === 0) {
if (b === 0) {
if (c === 0) {
resultDiv.innerHTML = "The equation is 0 = 0, which is true for all x (infinite solutions).";
} else {
resultDiv.innerHTML = "The equation is " + c + " = 0, which has no solution.";
}
} else {
var x = -c / b;
resultDiv.innerHTML = "This is a linear equation (a=0). The solution is x = " + x.toFixed(4) + "";
solutionSteps += "Step 2: Since 'a' is 0, this is a linear equation: " + b + "x + " + c + " = 0.";
solutionSteps += "To solve for x, rearrange the equation:";
solutionSteps += "" + b + "x = " + (-c) + "";
solutionSteps += "x = " + (-c) + " / " + b + "";
solutionSteps += "x = " + x.toFixed(4) + "";
}
resultDiv.innerHTML += solutionSteps;
return;
}
// Step 2: Calculate the Discriminant
var discriminant = (b * b) – (4 * a * c);
solutionSteps += "Step 2: Calculate the Discriminant (Δ).";
solutionSteps += "The discriminant is given by the formula: Δ = b² - 4ac";
solutionSteps += "Substitute the values: Δ = (" + b + ")² - 4 * (" + a + ") * (" + c + ")";
solutionSteps += "Δ = " + (b*b) + " - " + (4*a*c) + "";
solutionSteps += "Δ = " + discriminant.toFixed(4) + "";
var x1, x2;
var summary = "";
// Step 3: Determine the nature of the roots and calculate them using the quadratic formula
solutionSteps += "Step 3: Determine the nature of the roots and calculate them using the quadratic formula.";
solutionSteps += "The quadratic formula is: x = [-b ± sqrt(Δ)] / (2a)";
if (discriminant > 0) {
// Two distinct real roots
x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
summary = "The equation has two distinct real roots:";
summary += "x₁ = " + x1.toFixed(4) + "";
summary += "x₂ = " + x2.toFixed(4) + "";
solutionSteps += "Since Δ (" + discriminant.toFixed(4) + ") > 0, there are two distinct real roots.";
solutionSteps += "Calculate x₁: x₁ = [-" + b + " + sqrt(" + discriminant.toFixed(4) + ")] / (2 * " + a + ")";
solutionSteps += "x₁ = [" + (-b) + " + " + Math.sqrt(discriminant).toFixed(4) + "] / " + (2*a) + "";
solutionSteps += "x₁ = " + x1.toFixed(4) + "";
solutionSteps += "Calculate x₂: x₂ = [-" + b + " - sqrt(" + discriminant.toFixed(4) + ")] / (2 * " + a + ")";
solutionSteps += "x₂ = [" + (-b) + " - " + Math.sqrt(discriminant).toFixed(4) + "] / " + (2*a) + "";
solutionSteps += "x₂ = " + x2.toFixed(4) + "";
} else if (discriminant === 0) {
// One real root (repeated root)
x1 = -b / (2 * a);
summary = "The equation has one real (repeated) root:";
summary += "x = " + x1.toFixed(4) + "";
solutionSteps += "Since Δ (" + discriminant.toFixed(4) + ") = 0, there is one real (repeated) root.";
solutionSteps += "Calculate x: x = -" + b + " / (2 * " + a + ")";
solutionSteps += "x = " + (-b) + " / " + (2*a) + "";
solutionSteps += "x = " + x1.toFixed(4) + "";
} else {
// Two complex conjugate roots
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
summary = "The equation has two complex conjugate roots:";
summary += "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i";
summary += "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
solutionSteps += "Since Δ (" + discriminant.toFixed(4) + ") < 0, there are two complex conjugate roots.";
solutionSteps += "Calculate the real part: Real Part = -" + b + " / (2 * " + a + ") = " + realPart.toFixed(4) + "";
solutionSteps += "Calculate the imaginary part: Imaginary Part = sqrt(abs(" + discriminant.toFixed(4) + ")) / (2 * " + a + ") = " + imaginaryPart.toFixed(4) + "";
solutionSteps += "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i";
solutionSteps += "x₂ = " + realPart.toFixed(4) + " - " + imaginaryPart.toFixed(4) + "i";
}
resultDiv.innerHTML = summary + solutionSteps;
}
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Understanding the Quadratic Equation and Its Solutions
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where:
xrepresents the unknown variable.a,b, andcare coefficients, withanot equal to zero. Ifawere zero, the equation would become a linear equation (bx + c = 0).
The Quadratic Formula
The most common method to find the roots (solutions) of a quadratic equation is by using the quadratic formula. The roots are the values of x that satisfy the equation. The formula is:
x = [-b ± sqrt(b² - 4ac)] / (2a)
This formula provides two potential solutions due to the "±" (plus or minus) sign, which accounts for the two possible square roots of a number.
The Discriminant (Δ)
A crucial part of the quadratic formula is the expression under the square root sign: b² - 4ac. This is called the discriminant, often denoted by the Greek letter Delta (Δ). The value of the discriminant tells us about the nature of the roots without actually calculating them:
- If Δ > 0 (Discriminant is positive): The equation has two distinct real roots. This means there are two different numerical solutions for
x. Graphically, the parabola intersects the x-axis at two different points. - If Δ = 0 (Discriminant is zero): The equation has exactly one real root (also called a repeated or double root). This means both solutions from the quadratic formula are identical. Graphically, the parabola touches the x-axis at exactly one point (its vertex).
- If Δ < 0 (Discriminant is negative): The equation has two complex conjugate roots. This means there are no real number solutions for
x. Instead, the solutions involve imaginary numbers. Graphically, the parabola does not intersect the x-axis at all.
How to Use the Calculator
Our Quadratic Equation Solver simplifies the process of finding roots and understanding the solution steps:
- Identify Coefficients: Look at your quadratic equation and identify the values for
a,b, andc. Remember to include their signs (e.g., if you havex² - 3x + 2 = 0, thena=1,b=-3,c=2). - Enter Values: Input these values into the respective fields in the calculator.
- Calculate: Click the "Calculate Roots" button.
- Review Results: The calculator will display the roots (x₁ and x₂) and a detailed step-by-step breakdown of how those roots were derived, including the calculation of the discriminant and its interpretation.
Examples
Example 1: Two Distinct Real Roots
Consider the equation: x² - 5x + 6 = 0
- a = 1
- b = -5
- c = 6
Discriminant (Δ): (-5)² - 4 * 1 * 6 = 25 - 24 = 1
Since Δ > 0, there are two distinct real roots.
Roots:
x₁ = [5 + sqrt(1)] / (2 * 1) = (5 + 1) / 2 = 6 / 2 = 3x₂ = [5 - sqrt(1)] / (2 * 1) = (5 - 1) / 2 = 4 / 2 = 2
The roots are 3 and 2.
Example 2: One Real (Repeated) Root
Consider the equation: x² - 4x + 4 = 0
- a = 1
- b = -4
- c = 4
Discriminant (Δ): (-4)² - 4 * 1 * 4 = 16 - 16 = 0
Since Δ = 0, there is one real (repeated) root.
Root:
x = [4 ± sqrt(0)] / (2 * 1) = 4 / 2 = 2
The root is 2.
Example 3: Two Complex Conjugate Roots
Consider the equation: x² + 2x + 5 = 0
- a = 1
- b = 2
- c = 5
Discriminant (Δ): (2)² - 4 * 1 * 5 = 4 - 20 = -16
Since Δ < 0, there are two complex conjugate roots.
Roots:
x = [-2 ± sqrt(-16)] / (2 * 1) = [-2 ± 4i] / 2x₁ = -1 + 2ix₂ = -1 - 2i
The roots are -1 + 2i and -1 – 2i.