Enter the coefficients a, b, and c for your quadratic equation in the form ax² + bx + c = 0 to find its roots.
function calculateQuadraticRoots() {
var a = parseFloat(document.getElementById('coefficientA').value);
var b = parseFloat(document.getElementById('coefficientB').value);
var c = parseFloat(document.getElementById('coefficientC').value);
var resultDiv = document.getElementById('quadraticResult');
var output = ";
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.';
return;
}
if (a === 0) {
output += 'Step 1: Identify coefficients.';
output += 'a = ' + a + ', b = ' + b + ', c = ' + c + ";
output += 'Since the coefficient \'a\' is 0, this is not a quadratic equation. It\'s a linear equation: ' + b + 'x + ' + c + ' = 0.';
if (b === 0) {
if (c === 0) {
output += 'If b=0 and c=0, then 0=0, which is true for all x (infinite solutions).';
} else {
output += 'If b=0 and c≠0, then ' + c + ' = 0, which is false (no solution).';
}
} else {
var x_linear = -c / b;
output += 'Solving for x: ' + b + 'x = ' + (-c) + ";
output += 'x = ' + (-c) + ' / ' + b + ";
output += 'The solution for the linear equation is x = ' + x_linear.toFixed(4) + ";
}
resultDiv.innerHTML = output;
return;
}
output += 'Step 1: Identify coefficients.';
output += 'Given the equation ' + a + 'x² + ' + b + 'x + ' + c + ' = 0';
output += 'a = ' + a + ', b = ' + b + ', c = ' + c + ";
output += 'Step 2: Calculate the Discriminant (Δ).';
output += 'The discriminant is calculated using the formula: Δ = b² – 4ac';
var discriminant = (b * b) – (4 * a * c);
output += 'Δ = (' + b + ')² – 4(' + a + ')(' + c + ')';
output += 'Δ = ' + (b * b) + ' – ' + (4 * a * c) + ";
output += 'Δ = ' + discriminant.toFixed(4) + ";
output += 'Step 3: Determine the nature of the roots and apply the quadratic formula.';
output += 'The quadratic formula is: x = [-b ± √(Δ)] / 2a';
if (discriminant > 0) {
output += 'Since Δ (' + discriminant.toFixed(4) + ') > 0, there are two distinct real roots.';
var sqrtDiscriminant = Math.sqrt(discriminant);
var x1 = (-b + sqrtDiscriminant) / (2 * a);
var x2 = (-b – sqrtDiscriminant) / (2 * a);
output += 'x₁ = [-(' + b + ') + √(' + discriminant.toFixed(4) + ')] / [2(' + a + ')]';
output += 'x₁ = [' + (-b) + ' + ' + sqrtDiscriminant.toFixed(4) + '] / ' + (2 * a) + ";
output += 'x₁ = ' + ((-b + sqrtDiscriminant)).toFixed(4) + ' / ' + (2 * a) + ";
output += 'Root 1 (x₁) = ' + x1.toFixed(4) + ";
output += 'x₂ = [-(' + b + ') – √(' + discriminant.toFixed(4) + ')] / [2(' + a + ')]';
output += 'x₂ = [' + (-b) + ' – ' + sqrtDiscriminant.toFixed(4) + '] / ' + (2 * a) + ";
output += 'x₂ = ' + ((-b – sqrtDiscriminant)).toFixed(4) + ' / ' + (2 * a) + ";
output += 'Root 2 (x₂) = ' + x2.toFixed(4) + ";
} else if (discriminant === 0) {
output += 'Since Δ (' + discriminant.toFixed(4) + ') = 0, there is one real root (a repeated root).';
var x = -b / (2 * a);
output += 'x = [-(' + b + ') ± √(0)] / [2(' + a + ')]';
output += 'x = ' + (-b) + ' / ' + (2 * a) + ";
output += 'The single real root (x) = ' + x.toFixed(4) + ";
} else { // discriminant < 0
output += 'Since Δ (' + discriminant.toFixed(4) + ') < 0, there are two complex conjugate roots.';
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
output += 'x = [-(' + b + ') ± √(' + discriminant.toFixed(4) + ')] / [2(' + a + ')]';
output += 'x = [' + (-b) + ' ± i√(' + Math.abs(discriminant).toFixed(4) + ')] / ' + (2 * a) + '';
output += 'x = ' + realPart.toFixed(4) + ' ± i * ' + imaginaryPart.toFixed(4) + '';
output += 'Root 1 (x₁) = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i';
output += 'Root 2 (x₂) = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i';
}
resultDiv.innerHTML = output;
}
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Understanding Quadratic Equations and How to Solve Them
Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. One of the most common and important types of equations you'll encounter in algebra is the quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is squared but no term with a higher power.
What is a Quadratic Equation?
A standard quadratic equation is expressed in the form:
ax² + bx + c = 0
Where:
x represents the unknown variable.
a, b, and c are coefficients, which are known numerical values.
a cannot be equal to 0. If a were 0, the x² term would disappear, and the equation would become a linear equation (bx + c = 0).
The solutions to a quadratic equation are also known as its "roots" or "zeros." These are the values of x that make the equation true.
The Quadratic Formula: Your Key to Solutions
While some quadratic equations can be solved by factoring or completing the square, the most universal method is using the quadratic formula. This formula directly provides the roots of any quadratic equation, regardless of its complexity:
x = [-b ± √(b² - 4ac)] / 2a
Let's break down the components of this formula:
-b: The negative of the coefficient b.
±: This symbol indicates that there will generally be two solutions: one where you add the square root term and one where you subtract it.
√(b² - 4ac): This is the square root of the discriminant.
2a: Twice the coefficient a.
The Discriminant (Δ): Unveiling the Nature of Roots
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 values for x that satisfy the equation. Graphically, the parabola intersects the x-axis at two different points.
If Δ = 0 (Discriminant is zero): The equation has exactly one real root (sometimes 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 numbers that satisfy the equation. The solutions involve imaginary numbers. Graphically, the parabola does not intersect the x-axis at all.
How to Use the Quadratic Equation Solver
Our calculator simplifies the process of finding the roots of a quadratic equation. Here's how to use it:
Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, the equation must be in the form ax² + bx + c = 0.
Enter Values: Input these numerical values into the respective fields for 'Coefficient a', 'Coefficient b', and 'Coefficient c'.
Calculate: Click the "Calculate Roots" button.
Review Steps and Results: The calculator will display the step-by-step process, including the calculation of the discriminant and the final roots, along with their nature (real or complex).
Example Scenarios:
Let's look at a few examples to illustrate the different types of roots:
Example 1: Two Distinct Real Roots
Consider the equation: x² - 3x + 2 = 0
a = 1
b = -3
c = 2
Calculation:
Δ = (-3)² – 4(1)(2) = 9 – 8 = 1
Since Δ > 0, there are two distinct real roots.
x = [ -(-3) ± √(1) ] / 2(1)
x = [ 3 ± 1 ] / 2
x₁ = (3 + 1) / 2 = 4 / 2 = 2
x₂ = (3 – 1) / 2 = 2 / 2 = 1
The roots are x = 2 and x = 1.
Example 2: One Real Root (Repeated)
Consider the equation: x² - 4x + 4 = 0
a = 1
b = -4
c = 4
Calculation:
Δ = (-4)² – 4(1)(4) = 16 – 16 = 0
Since Δ = 0, there is one real root.
x = [ -(-4) ± √(0) ] / 2(1)
x = [ 4 ± 0 ] / 2
x = 4 / 2 = 2
The root is x = 2.
Example 3: Two Complex Conjugate Roots
Consider the equation: x² + 2x + 5 = 0
a = 1
b = 2
c = 5
Calculation:
Δ = (2)² – 4(1)(5) = 4 – 20 = -16
Since Δ < 0, there are two complex conjugate roots.
x = [ -(2) ± √(-16) ] / 2(1)
x = [ -2 ± 4i ] / 2
x₁ = -1 + 2i
x₂ = -1 – 2i
The roots are x = -1 + 2i and x = -1 - 2i.
Using this calculator, you can quickly and accurately solve any quadratic equation and understand the steps involved in arriving at the solution.