Contents

# Radical Equation Calculator + Online Solver With Free Steps

The **Radical Equation Calculator** solves a given radical equation for its roots and plots it. A radical equation is one with variables under the radical sign “$\surd\,$” as in:

\[ \text{radical equation} : \sqrt[n]{\text{variable terms}} + \text{other terms} = 0 \]

\[ \sqrt{5x^2+10x}+4x-7 = 0 \]

The calculator **supports multi-variable equations**, but the **intended usage is for single-variable ones**. That is because the calculator accepts only one equation at a time and cannot solve systems of simultaneous equations where we have n equations with m unknowns.

Thus, for multi-variable equations, the calculator outputs roots in terms of the other variables.

## What Is the Radical Equation Calculator?

**The Radical Equation Calculator is an online tool that evaluates the roots for a given radical equation representing a polynomial of any degree and plots the results. **

The **calculator interface** consists of a single text box labeled **“Equation.” **It is self-explanatory – you enter the radical equation to solve here. You may use any number of variables, but, as mentioned earlier, the intended usage is for single-variable polynomials of any degree.

## How To Use the Radical Equation Calculator?

You can use the **Radical Equation Calculator** by entering the given radical equation into the input text box. For example, suppose you want to solve the equation:

\[ 7x^5 +\sqrt{6x^3 + 3x^2}-2x-4 = 0 \]

Then you can use the calculator by following the step-by-step guidelines below.

### Step 1

Enter the equation in the text box. Enclose the radical term in “sqrt(radical term)” without quotes. In the example above, you would enter “7x^5+sqrt(6x^3+3x^2)-2x-4=0” without quotes.

**Note: Do not enter just the side of the equation with the polynomial!** Otherwise, the results will not contain the roots.

### Step 2

Press the **Submit **button to get the results.

### Results

The result section primarily consists of:

**Input:**The calculator’s interpretation of the input equation. Useful for verifying the equation and ensuring that the calculator handles it correctly.**Root Plots:**2D/3D plots with the roots highlighted. If at least one of the roots is complex, the calculator additionally draws them on the complex plane.**Roots/Solution:**These are the exact values of the roots. If they are a mixture of complex and real values, the calculator shows them in the separate sections**“Real Solutions”**and**“Complex Solutions.”**

There are also a couple of secondary sections (possibly more for different inputs):

**Number Line:**The real roots as they fall onto the number line.**Alternate Forms:**Various rearrangements of the input equation.

**For the example equation**, the calculator finds a mixture of real and complex roots:

\[ x_{r} \approx 0.858578 \]

\[ x_{c_1,\,c_2} \approx 0.12875 \pm 0.94078i \qquad x_{c_3,\,c_4} \approx -0.62771 \pm 0.41092i \]

## How Does the Radical Equation Calculator Work?

The **Radical Equation Calculator **works by isolating the radical term on one side of the equation and squaring both sides to **remove** the radical sign. After that, it brings all the variable and constant terms to one side of the equation, keeping 0 on the other end. Finally, it solves for the roots of the equation, which is now a standard polynomial of some degree d.

### Higher-order Polynomials

The calculator can quickly solve for polynomials with degrees greater than four. That is significant because there is no general formulation for solving d-degree polynomials with d > 4.

Extracting the roots of these higher-order polynomials requires a more advanced method such as the iterative **Newton** method. By hand, this method takes a long time because it is iterative, requires initial guesses, and may fail to converge for certain functions/guesses. However, this is not a problem for the calculator!

## Solved Examples

We will stick to lower-order polynomials in the following examples to explain the basic concept since solving higher-order polynomials with the Newton method will take a lot of time and space.

### Example 1

Consider the following equation:

\[ 11 + \sqrt{x-5} = 5 \]

Calculate the roots if possible. If not possible, explain why.

### Solution

Isolating the radical term:

\[ \begin{aligned} \sqrt{x-5} &= 5-11 \\ &= -6 \end{aligned} \]

Since the square root of a number cannot be negative, we can see that no solution exists for this equation. The calculator verifies this as well.

### Example 2

Solve the following equation for y in terms of x.

\[ \sqrt{5x+3y}-3 = 0 \]

### Solution

Isolating the radicals:

\[ \sqrt{5x+3y} = 3 \]

Since this is a positive number, we are safe to proceed. Squaring both sides of the equation:

\[ 5x+3y = 3^2 = 9 \]

Rearranging all terms to one side:

**5x+3y-9 = 0 **

It is the equation of a line! Solving for y:

**3y = -5x+9**

Dividing both sides by 3:

\[ y = -\frac{5}{3}x + 3 \]

The y-intercept of this line is at 3. Let us verify this on a graph:

The calculator also provides these results. Note that as we had only one equation, the solution is not a single point. It is constrained to a line instead. Similarly, if we had three variables instead, the set of possible solutions would lie on a plane!

### Example 3

Find the roots for the following equation:

\[ \sqrt{10x^2+20x}-3 = 0 \]

### Solution

Separating the radical term and squaring both sides after:

\[ \sqrt{10x^2 + 20x} = 3 \]

\[ 10x^2 + 20x = 9 \, \Rightarrow \, 10x^2+20x-9 = 0 \]

That is a quadratic equation in x. Using the quadratic formula with a = 10, b = 20, and c = -9:

\begin{align*} x_1,\, x_2 & = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\\\ & = \frac{-20 \pm \sqrt{20^2-4(10)(-9)}}{2(10)} \\\\ & = \frac{-20 \pm \sqrt{400+360}}{20} \\\\ & = \frac{-20 \pm \sqrt{760}}{20} \\\\ & = \frac{-20 \pm 27.5681}{20} \\\\ & = -1 \pm 1.3784 \end{align*}

We get the roots:

\[ \therefore , x_1 = 0.3784 \quad , \quad x_2 = -2.3784 \]

The calculator outputs the roots in their exact form:

\[ x_1 = -1 + \sqrt{\frac{19}{10}} \approx 0.3784 \quad,\quad x_2 = -1-\sqrt{\frac{19}{10}} \approx -2.3784 \]

The plot is below:

### Example 4

Consider the following radical with nested square roots:

\[ \sqrt{\sqrt{x^2-4x}-9x}-6 = 0 \]

Evaluate its roots.

### Solution

First, we isolate the outer radical as usual:

\[ \sqrt{\sqrt{x^2-4x}-9x} = 6 \]

Squaring both sides:

\[ \sqrt{x^2-4x}-9x = 36 \]

Now we need to remove the second radical sign as well, so we isolate the radical term again:

\[ \sqrt{x^2-4x} = 9x+36 \]

\[ x^2-4x = 81x^2+648x+1296 \]

\[ 80x^2+652x+1296 = 0 \]

Dividing both sides by 4:

\[ 20x^2+163x+324 = 0 \]

Solving using the quadratic formula with a = 20, b = 163, c = 324:

\begin{align*} x_1,\, x_2 & = \frac{-163 \pm \sqrt{163^2-4(20)(324)}}{2(20)} \\\\ & = \frac{-163 \pm \sqrt{26569 – 25920}}{40} \\\\ &= \frac{-163 \pm \sqrt{649}}{40} \\\\ & = \frac{-163 \pm 25.4755}{40} \\\\ & = -4.075 \pm 0.63689 \end{align*}

\[ \therefore \,\,\, x_1 = -3.4381 \quad , \quad x_2 = -4.7119 \]

However, if we plug in $x_2$ = -4.7119 into our original equation, the two sides are not equal:

\[ 6.9867-6 \neq 0 \]

Whereas with $x_1$ = -3.4381, we get:

\[ 6.04-6 \approx 0 \]

The slight error is due to the decimal approximation. We can verify this in the figure as well:

*All graphs/images were created with GeoGebra.*