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(function(){
“use strict”;function startRadicalSolver(){
var root=document.getElementById(“cd-radical-equation-solver”);
if(!root){return;}var ids=[“a”,”b”,”c”,”d”,”e”,”f”];
var inputs={};
ids.forEach(function(id){
inputs[id]=document.getElementById(“cd-rad-“+id);
});var precisionEl=document.getElementById(“cd-rad-precision”);
var previewEl=document.getElementById(“cd-rad-preview-text”);
var solveBtn=document.getElementById(“cd-rad-solve”);
var copyBtn=document.getElementById(“cd-rad-copy”);
var clearBtn=document.getElementById(“cd-rad-clear”);
var errorEl=document.getElementById(“cd-rad-error”);
var resultEl=document.getElementById(“cd-rad-result”);
var mainResultEl=document.getElementById(“cd-rad-main-result”);
var resultNoteEl=document.getElementById(“cd-rad-result-note”);
var candidatesEl=document.getElementById(“cd-rad-candidates”);
var stepsEl=document.getElementById(“cd-rad-steps”);
var copyStatusEl=document.getElementById(“cd-rad-copy-status”);
var presetBtns=root.querySelectorAll(“.cd-rad-preset”);
var lastCopyText=””;function nearZero(n){
return Math.abs(n)=1000000000000){
return n.toExponential(Math.min(p,8));
}
if(abs>0){
if(abs0){
out+=” + “+fmt(c,p);
}else{
out+=” – “+fmt(Math.abs(c),p);
}
}
if(out===””){out=”0″;}
return out;
}function formatRadicalPart(a,b,c,p){
var inside=formatLinear(b,c,p);
var rad=”sqrt(“+inside+”)”;
if(nearZero(a-1)){return rad;}
if(nearZero(a+1)){return “-“+rad;}
return fmt(a,p)+”*”+rad;
}function addConstant(base,c,p){
var out=base;
if(!nearZero(c)){
if(c>0){
out+=” + “+fmt(c,p);
}else{
out+=” – “+fmt(Math.abs(c),p);
}
}
return out;
}function equationText(v,p){
var left=addConstant(formatRadicalPart(v.a,v.b,v.c,p),v.d,p);
var right=formatLinear(v.e,v.f,p);
return left+” = “+right;
}function formatPolynomial(a,b,c,p){
var parts=[];if(!nearZero(a)){
if(nearZero(a-1)){
parts.push(“x^2”);
}else if(nearZero(a+1)){
parts.push(“-x^2″);
}else{
parts.push(fmt(a,p)+”x^2″);
}
}if(!nearZero(b)){
var absB=Math.abs(b);
var body=nearZero(absB-1)?”x”:fmt(absB,p)+”x”;
if(parts.length===0){
parts.push(b<0?"-"+body:body);
}else{
parts.push(b<0?"- "+body:"+ "+body);
}
}if(!nearZero(c)){
var absC=fmt(Math.abs(c),p);
if(parts.length===0){
parts.push(c<0?"-"+absC:absC);
}else{
parts.push(c<0?"- "+absC:"+ "+absC);
}
}if(parts.length===0){parts.push("0");}
return parts.join(" ")+" = 0";
}function readValues(){
var values={};
var ok=true;
var message="";ids.forEach(function(id){
if(!ok){return;}
var raw=inputs[id].value.trim();
if(raw===""){
ok=false;
message="Enter a value for every coefficient.";
return;
}
var n=Number(raw);
if(!Number.isFinite(n)){
ok=false;
message="All coefficients must be valid finite numbers.";
return;
}
values[id]=n;
});if(ok){
if(nearZero(values.a)){
ok=false;
message="A cannot be 0 because the radical term would disappear.";
}
}if(ok){
if(nearZero(values.b)){
ok=false;
message="B cannot be 0 because x must appear inside the square root.";
}
}return {ok:ok,message:message,values:values};
}function showError(message){
errorEl.textContent=message;
errorEl.style.display="block";
}function hideError(){
errorEl.textContent="";
errorEl.style.display="none";
}function updatePreview(){
var read=readValues();
if(read.ok){
previewEl.textContent=equationText(read.values,4);
}else{
previewEl.textContent="Enter valid coefficients to preview the equation.";
}
}function addStep(text){
var li=document.createElement("li");
li.textContent=text;
stepsEl.appendChild(li);
}function addCandidateChip(text,isValid){
var span=document.createElement("span");
span.className="cd-rad-chip "+(isValid?"cd-rad-valid":"cd-rad-invalid");
span.textContent=text;
candidatesEl.appendChild(span);
}function addUniqueCandidate(arr,x){
if(!Number.isFinite(x)){return;}
var duplicate=false;
arr.forEach(function(existing){
var scale=1+Math.abs(existing)+Math.abs(x);
if(Math.abs(existing-x)0){
return “x >= “+fmt(boundary,p);
}
return “x <= "+fmt(boundary,p);
}function solve(){
try{
hideError();
copyStatusEl.textContent="";var read=readValues();
if(!read.ok){
showError(read.message);
return;
}var v=read.values;
var p=Number(precisionEl.value);
var k=v.f-v.d;
var a2=v.a*v.a;
var qa=v.e*v.e;
var qb=2*v.e*k-a2*v.b;
var qc=k*k-a2*v.c;var derivedFinite=[qa,qb,qc,a2].every(Number.isFinite);
if(!derivedFinite){
showError("The coefficients are too large for a reliable calculation. Use smaller values.");
return;
}var candidates=[];
var solvingType="quadratic";if(nearZero(qa)){
solvingType="linear";
if(nearZero(qb)){
showError("The transformed equation cannot be solved reliably with these coefficients.");
return;
}
addUniqueCandidate(candidates,-qc/qb);
}else{
var disc=qb*qb-4*qa*qc;
if(!Number.isFinite(disc)){
showError("The calculation exceeded the supported numeric range. Use smaller coefficient values.");
return;
}var discScale=1+Math.abs(qb*qb)+Math.abs(4*qa*qc);
if(disc<0){
if(Math.abs(disc)=0){
var sqrtDisc=Math.sqrt(disc);if(nearZero(disc)){
addUniqueCandidate(candidates,-qb/(2*qa));
}else{
var q;
if(qb>=0){
q=-0.5*(qb+sqrtDisc);
}else{
q=-0.5*(qb-sqrtDisc);
}if(!nearZero(q)){
addUniqueCandidate(candidates,q/qa);
addUniqueCandidate(candidates,qc/q);
}else{
addUniqueCandidate(candidates,(-qb+sqrtDisc)/(2*qa));
addUniqueCandidate(candidates,(-qb-sqrtDisc)/(2*qa));
}
}
}
}var valid=[];
var rejected=[];
var checks=[];
var tolerance=1e-8;candidates.forEach(function(x){
var rad=v.b*x+v.c;
var scale=1+Math.abs(v.b*x)+Math.abs(v.c);
var domainTolerance=tolerance*scale;
var domainOk=rad>=-domainTolerance;if(!domainOk){
rejected.push(x);
checks.push({x:x,ok:false,reason:”outside the real square-root domain”});
return;
}var safeRad=Math.max(0,rad);
var lhs=v.a*Math.sqrt(safeRad)+v.d;
var rhs=v.e*x+v.f;
var diff=Math.abs(lhs-rhs);
var eqTolerance=tolerance*(1+Math.abs(lhs)+Math.abs(rhs));if(diff0){
resultNoteEl.textContent=”At least one algebraic candidate was rejected after checking the original equation.”;
}else{
resultNoteEl.textContent=”The candidate satisfies the square-root domain and the original equation.”;
}
}else{
var shown=valid.map(function(x){return fmt(x,p);});
mainResultEl.textContent=”Solutions: x = “+shown.join(“, “);
if(rejected.length>0){
resultNoteEl.textContent=”Valid solutions are shown above. Other algebraic candidates were rejected.”;
}else{
resultNoteEl.textContent=”Both candidates satisfy the original radical equation.”;
}
}checks.forEach(function(item){
if(item.ok){
addCandidateChip(“x = “+fmt(item.x,p)+” – valid”,true);
}else{
addCandidateChip(“x = “+fmt(item.x,p)+” – extraneous”,false);
}
});addStep(“Original equation: “+equationText(v,p));
addStep(“Real square-root domain: “+getDomainText(v,p)+”.”);
addStep(“Isolate the radical: “+formatRadicalPart(v.a,v.b,v.c,p)+” = “+formatLinear(v.e,k,p)+”.”);
addStep(“Square both sides: (“+fmt(v.a,p)+”)^2*(“+formatLinear(v.b,v.c,p)+”) = (“+formatLinear(v.e,k,p)+”)^2.”);
addStep(“After expansion: “+formatPolynomial(qa,qb,qc,p));if(candidates.length===0){
addStep(“The “+solvingType+” equation has no real candidate roots.”);
}else{
var candidateText=candidates.map(function(x){return fmt(x,p);}).join(“, “);
addStep(“Algebraic candidate values: x = “+candidateText+”.”);
checks.forEach(function(item){
if(item.ok){
addStep(“Check x = “+fmt(item.x,p)+”: left side = “+fmt(item.lhs,p)+”, right side = “+fmt(item.rhs,p)+”. The candidate is valid.”);
}else{
if(typeof item.lhs===”number”){
addStep(“Check x = “+fmt(item.x,p)+”: left side = “+fmt(item.lhs,p)+”, right side = “+fmt(item.rhs,p)+”. The candidate is extraneous.”);
}else{
addStep(“Check x = “+fmt(item.x,p)+”: rejected because it is outside the real square-root domain.”);
}
}
});
}var copyParts=[];
copyParts.push(“Equation: “+equationText(v,p));
copyParts.push(mainResultEl.textContent);
checks.forEach(function(item){
copyParts.push(“x = “+fmt(item.x,p)+”: “+(item.ok?”valid”:”extraneous”));
});
lastCopyText=copyParts.join(“\n”);resultEl.style.display=”block”;
}catch(err){
showError(“The calculator could not complete this input. Check the values and try again.”);
}
}function copyResult(){
copyStatusEl.textContent=””;
if(lastCopyText===””){
copyStatusEl.textContent=”Solve an equation first.”;
return;
}if(navigator.clipboard){
navigator.clipboard.writeText(lastCopyText).then(function(){
copyStatusEl.textContent=”Result copied.”;
}).catch(function(){
fallbackCopy();
});
}else{
fallbackCopy();
}
}function fallbackCopy(){
var area=document.createElement(“textarea”);
area.value=lastCopyText;
area.setAttribute(“readonly”,”readonly”);
area.style.position=”fixed”;
area.style.opacity=”0″;
document.body.appendChild(area);
area.select();
try{
document.execCommand(“copy”);
copyStatusEl.textContent=”Result copied.”;
}catch(err){
copyStatusEl.textContent=”Copy is not available in this browser.”;
}
document.body.removeChild(area);
}function clearAll(){
ids.forEach(function(id){
inputs[id].value=””;
});
precisionEl.value=”6″;
hideError();
previewEl.textContent=”Enter valid coefficients to preview the equation.”;
mainResultEl.textContent=”Enter coefficients and solve the equation.”;
resultNoteEl.textContent=””;
candidatesEl.innerHTML=””;
stepsEl.innerHTML=””;
lastCopyText=””;
copyStatusEl.textContent=””;
resultEl.style.display=”block”;
}function applyPreset(name){
var data;
if(name===”one”){
data={a:1,b:3,c:10,d:0,e:1,f:2};
}else if(name===”two”){
data={a:1,b:5,c:-4,d:-9,e:0,f:0};
}else{
data={a:2,b:1,c:3,d:1,e:1,f:0};
}ids.forEach(function(id){
inputs[id].value=data[id];
});
updatePreview();
solve();
}ids.forEach(function(id){
inputs[id].addEventListener(“input”,updatePreview);
inputs[id].addEventListener(“keydown”,function(event){
if(event.key===”Enter”){
solve();
}
});
});precisionEl.addEventListener(“change”,function(){
updatePreview();
});solveBtn.addEventListener(“click”,solve);
copyBtn.addEventListener(“click”,copyResult);
clearBtn.addEventListener(“click”,clearAll);presetBtns.forEach(function(btn){
btn.addEventListener(“click”,function(){
applyPreset(btn.getAttribute(“data-preset”));
});
});updatePreview();
solve();
}if(document.readyState===”loading”){
document.addEventListener(“DOMContentLoaded”,startRadicalSolver);
}else{
startRadicalSolver();
}
})();
Solve a Radical Equation with Steps
Use this radical equation solver for equations in the form A√(Bx + C) + D = Ex + F. It finds real candidate roots, checks the square-root domain, and substitutes every candidate back into the original equation to remove extraneous solutions.
Algebra
A cannot be 0.
B cannot be 0.
2
4
6
8
10
Equation: sqrt(3x + 10) = x + 2
Real solution
Solution: x = 2
The other algebraic candidate is extraneous and does not satisfy the original radical equation.
What the Solver Checks
Solving a square-root equation requires more than solving the equation produced after squaring. Squaring can create an extraneous solution: a number that satisfies the transformed equation but not the original one.
This calculator checks each candidate against both the square-root domain and the original equation. Only candidates that pass the substitution check appear as real solutions.
Method Used by the Radical Equation Solver
The supported equation is:
A√(Bx + C) + D = Ex + FFirst, isolate the radical term:
A√(Bx + C) = Ex + F − DThen square both sides:
A²(Bx + C) = (Ex + F − D)²After expanding and moving everything to one side, the equation becomes:
E²x² + [2E(F − D) − A²B]x + [(F − D)² − A²C] = 0If E = 0, the squared equation reduces to a linear equation. Otherwise, it is usually quadratic. Every resulting value must still be tested in the original radical equation.
Square-Root Domain
The expression under a real square root cannot be negative:
Bx + C ≥ 0The solver calculates this restriction automatically before accepting a solution.
How to Use the Calculator
- Match your equation to A√(Bx + C) + D = Ex + F.
- Enter the six coefficients. A coefficient that does not appear in your equation is usually entered as 0.
- Keep A and B nonzero so the equation contains a variable square-root term.
- Select the number of decimal places you want to display.
- Press Solve Equation.
- Read the candidate check and solution steps to see whether any algebraic root was rejected as extraneous.
Worked Example: √(3x + 10) = x + 2
For this equation:
A = 1, B = 3, C = 10, D = 0, E = 1, F = 2Square both sides:
3x + 10 = (x + 2)²Expand the right side and move all terms to one side:
3x + 10 = x² + 4x + 4 x² + x − 6 = 0Factor:
(x + 3)(x − 2) = 0The algebraic candidates are x = −3 and x = 2.
Check x = −3:
√(3(−3) + 10) = −3 + 2 → 1 ≠ −1So x = −3 is extraneous.
Check x = 2:
√(3(2) + 10) = 2 + 2 → 4 = 4The real solution is x = 2.
Why Extraneous Solutions Appear
Squaring does not preserve every restriction of the original equation. For example, the equations 1 = −1 and 1 = 1 are different, but squaring both sides of either equation produces 1 = 1.
A square root written with the radical symbol represents the principal, nonnegative square root. Because the squaring step can hide a sign mismatch, substitution into the original equation is part of solving a radical equation correctly.
Common Input Mistakes
| Issue | What happens | What to do |
|---|---|---|
| A = 0 | The square-root term disappears. | Enter the actual nonzero multiplier of the radical. |
| B = 0 | The variable is no longer inside the radical. | Use a nonzero x coefficient inside the square root. |
| Ignoring the domain | A candidate may make Bx + C negative. | Check the domain shown in the solution steps. |
| Accepting every root after squaring | An extraneous root may be reported as a solution. | Verify every candidate in the original equation. |
| Entering signs incorrectly | The calculator solves a different equation. | Enter negative coefficients directly, such as −4 for C in 5x − 4. |
What This Solver Supports
This calculator handles real equations containing one square-root term with a linear expression inside the radical and a linear expression on the other side:
A√(Bx + C) + D = Ex + FIt does not solve equations with two separate radicals, nested radicals, cube roots, fourth roots, variables in denominators, or nonlinear expressions such as √(x² + 1). Those forms can require repeated radical elimination or other algebraic methods.
Decimal results are rounded only for display. Candidate verification uses unrounded JavaScript numeric values before the result is shown.
Frequently Asked Questions
What is a radical equation?
A radical equation contains a variable inside a radical expression. An example is √(3x + 10) = x + 2.
Why must radical equation answers be checked?
Raising both sides to an even power can introduce values that satisfy the transformed equation but fail in the original equation. These are called extraneous solutions.
Can a radical equation have no real solution?
Yes. The transformed equation may have no real roots, every candidate may violate the square-root domain, or all candidates may fail when substituted into the original equation.
Can this calculator return two real solutions?
Yes. Squaring can produce a quadratic equation with two candidates, and both are kept if both satisfy the original radical equation.
Does this solver handle negative numbers inside a square root?
It works with real numbers only. A candidate is rejected if it makes the radicand Bx + C negative.
Why does squaring a radical equation often create a quadratic?
When the right side contains a linear expression such as Ex + F, squaring it produces an x² term. That turns the transformed equation into a quadratic unless the x coefficient on the right side is zero.
