In mathematics, the solution set is the collection of all values that satisfy a given equation or inequality. When you plug any value from the solution set back into the original equation, the equation holds true.
Linear Equations (ax + b = c)
A linear equation represents a straight line. For most linear equations, there is exactly one value in the solution set. The formula used to find this value is:
x = (c - b) / a
Example: If you have 2x + 4 = 12, the solution set is {4} because 2(4) + 4 = 12.
Quadratic Equations (ax² + bx + c = 0)
Quadratic equations involve a squared variable and can have zero, one, or two real solutions. We determine the solution set using the quadratic formula:
x = [-b ± sqrt(b² - 4ac)] / 2a
The "Discriminant" (b² – 4ac) tells us the nature of the solution set:
Positive: Two distinct real solutions.
Zero: One real solution (a repeated root).
Negative: No real solutions (the solution set contains complex or imaginary numbers).
Practical Applications
Solution sets are fundamental in fields ranging from physics (calculating the trajectory of a projectile) to economics (finding the break-even point where costs equal revenue). Using a solution set calculator streamlines these calculations, ensuring accuracy in algebraic manipulation.
function calculateLinear() {
var a = parseFloat(document.getElementById('linear_a').value);
var b = parseFloat(document.getElementById('linear_b').value);
var c = parseFloat(document.getElementById('linear_c').value);
var resultDiv = document.getElementById('linear_result');
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.style.display = 'block';
resultDiv.innerHTML = 'Please enter valid numerical values.';
return;
}
if (a === 0) {
if (b === c) {
resultDiv.innerHTML = 'Solution Set: { All Real Numbers } (Identity Equation)';
} else {
resultDiv.innerHTML = 'Solution Set: { Ø } (No Solution)';
}
} else {
var x = (c – b) / a;
resultDiv.innerHTML = 'Solution Set: { ' + x.toFixed(4).replace(/\.?0+$/, "") + ' }';
}
resultDiv.style.display = 'block';
}
function calculateQuadratic() {
var a = parseFloat(document.getElementById('quad_a').value);
var b = parseFloat(document.getElementById('quad_b').value);
var c = parseFloat(document.getElementById('quad_c').value);
var resultDiv = document.getElementById('quad_result');
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.style.display = 'block';
resultDiv.innerHTML = 'Please enter valid numerical values.';
return;
}
if (a === 0) {
resultDiv.style.display = 'block';
resultDiv.innerHTML = 'Value "a" cannot be 0 for a quadratic equation. Please use the linear solver.';
return;
}
var discriminant = (b * b) – (4 * a * c);
resultDiv.style.display = 'block';
if (discriminant > 0) {
var x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
resultDiv.innerHTML = 'Solution Set: { ' + x1.toFixed(4).replace(/\.?0+$/, "") + ', ' + x2.toFixed(4).replace(/\.?0+$/, "") + ' }';
} else if (discriminant === 0) {
var x = -b / (2 * a);
resultDiv.innerHTML = 'Solution Set: { ' + x.toFixed(4).replace(/\.?0+$/, "") + ' } (Repeated Root)';
} else {
var realPart = (-b / (2 * a)).toFixed(4).replace(/\.?0+$/, "");
var imagPart = (Math.sqrt(-discriminant) / (2 * a)).toFixed(4).replace(/\.?0+$/, "");
resultDiv.innerHTML = 'Solution Set (Complex): { ' + realPart + ' + ' + imagPart + 'i, ' + realPart + ' – ' + imagPart + 'i }';
}
}