Enter the coefficients for a radical equation of the form: n√( ax + b) = c
or
a + n√( bx + c) = d
Solution(s) will appear here.
Understanding and Solving Radical Equations
A radical equation is an equation that contains a variable within a radical symbol (like a square root, cube root, etc.). These equations are fundamental in algebra and appear in various mathematical and scientific contexts. The most common form involves square roots, but they can involve any nth root.
The General Form
While radical equations can take many forms, a common structure we can solve with this calculator is:
n√( Bx + C) + A = D
Where:
n√ represents the nth root (e.g., n=2 for square root, n=3 for cube root).
'x' is the variable we want to solve for.
B is the coefficient of 'x' inside the radical.
C is the constant term inside the radical.
A is a constant term outside the radical (added or subtracted).
D is the value the expression equals.
For simplicity, this calculator addresses equations where A (the 'coeffA' input) might be 0 or 1, and the form can be expressed as:
n√( bx + c) = d
(This is equivalent to the first form if A=0, or if A=1 and D=1, etc. We will adapt the inputs to fit this simplified form for the calculation logic, effectively treating the 'coeffA' as 0 unless it's explicitly 1 and part of the isolated radical term).
Steps to Solve a Radical Equation
Isolate the Radical: Get the radical term by itself on one side of the equation.
Eliminate the Radical: Raise both sides of the equation to the power of the root index (n). For a square root (n=2), you square both sides. For a cube root (n=3), you cube both sides, and so on.
Solve for x: Solve the resulting (usually linear or polynomial) equation for the variable 'x'.
Check for Extraneous Solutions: This is a CRITICAL step. Substitute the solution(s) back into the *original* radical equation. Any solution that makes the original equation false (e.g., results in taking an even root of a negative number or leads to an inequality) is an extraneous solution and must be discarded.
How This Calculator Works
This calculator simplifies the process for equations that can be rearranged into the form:
n√( bx + c) = d
It performs the following operations:
It takes the user-provided Root Index (n).
It takes the coefficients b (Coefficient 'b'), c (Constant 'c' inside root), and d (Right Side Value 'd').
It calculates dn.
It sets up the equation: bx + c = dn.
It solves for x: bx = dn – c, leading to x = (dn – c) / b.
Important Note: This calculator primarily solves for cases where the radical is already isolated. If your equation is in the form n√( bx + c) + A = D, you first need to rearrange it to n√( bx + c) = D – A. Then, you would use D – A as the 'd' value in the calculator. The 'coeffA' input is generally ignored in the direct calculation here but is included for conceptual clarity of the general form. A coefficient of 'b' equal to 0 will result in an error or no solution unless dn – c is also 0.
Even Roots and Negative Numbers: Be mindful that for even roots (like square roots, 4th roots), the expression inside the radical (bx + c) must be non-negative. This calculator does not explicitly check for this condition *before* calculation but it's crucial for verifying solutions.
Check: √(2 * 11 + 3) = √(22 + 3) = √25 = 5. The solution is valid.
Using the calculator with these inputs should yield x = 11.
Another Example
Solve: 3√(x – 7) = 2
Root Index (n): 3
Coefficient 'b': 1
Constant 'c' (inside root): -7
Right Side Value 'd': 2
Calculation:
Isolate radical: Done.
Eliminate radical: Cube both sides. (3√(x – 7))3 = 23 => x – 7 = 8
Solve for x: x = 8 + 7 => x = 15
Check: 3√(15 – 7) = 3√8 = 2. The solution is valid.
The calculator should output x = 15.
Edge Cases
Coefficient 'b' is 0: If b=0, the equation becomes n√c = d. This will only have a solution if c = dn. If c ≠ dn, there is no solution. If c = dn, any 'x' is technically a solution, but this calculator assumes b≠0 for a unique x.
Even Root of Negative Result: If you are solving a square root equation and after isolating the radical, the right side 'd' is negative (e.g., √(expression) = -3), there is no real solution, as even roots cannot produce negative real numbers. This calculator might produce a result based on dn, but it should be flagged as potentially extraneous.
No Solution: Sometimes, after performing the steps, the derived solution(s) do not satisfy the original equation. Always check your answers!
function solveRadicalEquation() {
var n = parseFloat(document.getElementById("rootIndex").value);
var b = parseFloat(document.getElementById("coeffB").value);
var c = parseFloat(document.getElementById("constantC_inside").value);
var d = parseFloat(document.getElementById("equalSideD").value);
var resultDiv = document.getElementById("result");
// Clear previous results and styling
resultDiv.innerHTML = "Solution(s) will appear here.";
resultDiv.style.color = "#004a99";
resultDiv.style.backgroundColor = "#e7f3ff";
resultDiv.style.borderColor = "#004a99";
// Input Validation
var inputsValid = !isNaN(n) && !isNaN(b) && !isNaN(c) && !isNaN(d);
if (!inputsValid) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all fields.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
}
if (n <= 0 || !Number.isInteger(n)) {
resultDiv.innerHTML = "Error: Root index (n) must be a positive integer.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
}
// Handle the case where the radical is not isolated, e.g., A + n√(bx+c) = D
// For simplicity, we assume the equation is already in the form n√(bx+c) = d
// If coeffA was intended to be subtracted from D, the user should input D – A as 'd'.
// If coeffA is non-zero and not handled by the user's input for 'd', it's ignored in this simplified solver.
if (b === 0) {
// Equation simplifies to n√c = d
var c_raised = Math.pow(c, 1/n); // Calculate nth root of c
var d_raised = Math.pow(d, n); // Calculate d to the power of n
// Check if n is even and d is negative
if (n % 2 === 0 && d < 0) {
resultDiv.innerHTML = "No Real Solution: An even root cannot equal a negative number.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
}
// Check if n is even and c is negative
if (n % 2 === 0 && c < 0) {
resultDiv.innerHTML = "No Real Solution: Cannot take an even root of a negative number inside the radical.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
}
var nth_root_of_c;
// Handle potential issues with Math.pow for negative bases and fractional exponents
if (c < 0 && n % 2 === 0) {
// This case should ideally be caught earlier, but as a safeguard
resultDiv.innerHTML = "No Real Solution: Cannot compute even root of negative number.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
} else if (c < 0 && n % 2 !== 0) {
nth_root_of_c = -Math.pow(-c, 1/n);
} else {
nth_root_of_c = Math.pow(c, 1/n);
}
if (Math.abs(nth_root_of_c – d) < 1e-9) { // Using tolerance for float comparison
resultDiv.innerHTML = "Infinite Solutions: Equation simplifies to a true statement (n√c = d).";
resultDiv.style.color = "#007bff";
resultDiv.style.backgroundColor = "#cfe2ff";
resultDiv.style.borderColor = "#9ec5fe";
} else {
resultDiv.innerHTML = "No Solution: Equation simplifies to a false statement (n√c ≠ d).";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
}
return;
}
// Calculate d^n
var d_to_the_n = Math.pow(d, n);
// Check for even roots and negative 'd' which leads to no real solution
if (n % 2 === 0 && d < 0) {
resultDiv.innerHTML = "No Real Solution: An even root cannot equal a negative number.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
resultDiv.style.borderColor = "#f5c6cb";
return;
}
// Calculate x = (d^n – c) / b
var numerator = d_to_the_n – c;
var x = numerator / b;
// Check if the calculated x results in a non-negative value under an even root
var value_under_radical = b * x + c;
if (n % 2 === 0 && value_under_radical < 0) {
resultDiv.innerHTML = "Extraneous Solution: Calculated x = " + x.toFixed(4) + " leads to a negative under an even root.";
resultDiv.style.color = "#ffc107"; // Warning color
resultDiv.style.backgroundColor = "#fff3cd";
resultDiv.style.borderColor = "#ffeeba";
return;
}
// Display the solution
resultDiv.innerHTML = "x = " + x.toFixed(4);
resultDiv.style.color = "#155724"; // Success green text
resultDiv.style.backgroundColor = "#d4edda"; // Success green background
resultDiv.style.borderColor = "#c3e6cb"; // Success green border
}