Quadratic Formula Solver
Enter the coefficients a, b, and c for a quadratic equation in the form ax² + bx + c = 0 to find its roots.
0 = 0, which has infinitely many solutions.";
} else {
resultDiv.innerHTML = "This is the equation " + c + " = 0, which has no solution.";
}
} else {
// Linear equation: bx + c = 0
var x = -c / b;
resultDiv.innerHTML = "This is a linear equation (a=0). The solution is: x = " + x.toFixed(4) + "";
}
return;
}
var discriminant = b * b – 4 * a * c;
if (discriminant > 0) {
var x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
resultDiv.innerHTML = "The equation has two distinct real roots:" +
"x₁ = " + x1.toFixed(4) + "" +
"x₂ = " + x2.toFixed(4) + "";
} else if (discriminant === 0) {
var x = -b / (2 * a);
resultDiv.innerHTML = "The equation has one real root (or two equal real roots):" +
"x = " + x.toFixed(4) + "";
} else { // discriminant < 0
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
resultDiv.innerHTML = "The equation has two complex conjugate roots:" +
"x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i" +
"x₂ = " + realPart.toFixed(4) + " - " + imaginaryPart.toFixed(4) + "i";
}
}
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text-align: center;
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color: #333;
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width: 100%;
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margin-top: 20px;
transition: background-color 0.3s ease;
}
.calculator-container button:hover {
background-color: #0056b3;
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.calc-result-area {
margin-top: 25px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #e9ecef;
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Understanding the Quadratic Formula in Pre-Calculus
Pre-calculus serves as a bridge between algebra and calculus, introducing more advanced concepts and problem-solving techniques. One of the most fundamental and frequently used tools in pre-calculus (and beyond) is the quadratic formula. This formula provides a direct method for finding the roots (or solutions) of any quadratic equation.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which 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, andcare real numbers.a ≠ 0(Ifawere 0, the equation would become linear:bx + c = 0).xis the variable for which we are solving.
Quadratic equations are used to model various real-world phenomena, from the trajectory of a projectile to the shape of a parabolic antenna.
The Quadratic Formula Explained
The quadratic formula is derived by completing the square on the standard form of a quadratic equation. It states that the solutions for x are given by:
x = [-b ± √(b² - 4ac)] / 2a
This formula allows you to find the values of x that satisfy the equation, regardless of whether the roots are real or complex.
The Discriminant: Unveiling the Nature of Roots
A crucial part of the quadratic formula is the expression under the square root sign: b² - 4ac. This term 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 of the quadratic equation:
- If
Δ > 0(Discriminant is positive): The equation has two distinct real roots. This means the parabola (the graph of the quadratic function) 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). In this case, 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 the parabola does not intersect the x-axis at all. The roots involve the imaginary uniti, wherei = √(-1).
How to Use the Quadratic Formula Solver
Our Quadratic Formula Solver simplifies the process of finding roots. Simply input the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0 into the respective fields. The calculator will then apply the quadratic formula and display the roots, indicating whether they are real or complex.
Examples:
-
Equation:
x² - 3x + 2 = 0a = 1b = -3c = 2
Discriminant:
(-3)² - 4(1)(2) = 9 - 8 = 1(Positive)Roots:
x₁ = 2,x₂ = 1(Two distinct real roots) -
Equation:
x² + 4x + 4 = 0a = 1b = 4c = 4
Discriminant:
(4)² - 4(1)(4) = 16 - 16 = 0(Zero)Root:
x = -2(One real root) -
Equation:
x² + x + 1 = 0a = 1b = 1c = 1
Discriminant:
(1)² - 4(1)(1) = 1 - 4 = -3(Negative)Roots:
x₁ = -0.5 + 0.8660i,x₂ = -0.5 - 0.8660i(Two complex conjugate roots) -
Equation:
0x² + 5x - 10 = 0(Linear Equation)a = 0b = 5c = -10
The calculator will identify this as a linear equation and solve for
x:5x - 10 = 0, so5x = 10,x = 2.
Mastering the quadratic formula is a key skill in pre-calculus, essential for solving equations, analyzing functions, and preparing for more advanced mathematical topics.