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# Root Finder Calculator + Online Solver With Free Steps

The root finder calculator is used to **find the roots of a polynomial** of any degree greater than zero. The **number of roots** of the equation depends upon the **degree of the polynomial**.

This calculator takes the polynomial equation as input and provides all the possible solutions to the equation and **plots** **the solution in a 2-D** **plane**.

## What Is a Root Finder Calculator?

**A Root Finder Calculator is an online calculator that computes the roots or solutions of a function of nth degree where n = 1,2,3,4 and so on.**

To explain its functioning, consider a **quadratic function** which is a **second-degree polynomial** written in the form \[ (p)x^2 + (q)x + r = 0 \] where p and $q$ are coefficients of (x)^2 and x, respectively, and r is a constant. If p = 0, the function becomes** linear**.

The roots of a quadratic equation are the **x-intercepts** of the function. The x-intercepts are obtained by putting the function y = f(x) = 0.

These points lie on the x-axis, giving the solutions of the function. This calculator can also find the x-intercepts of any polynomial with both real and imaginary roots.

## How To Use the Root Finder Calculator

Here are the steps required to use the root finder calculator.

### Step 1:

The calculator shows a quadratic equation of the form:

\[ (p)x^2 + (q)x + r = 0 \]

with p = 1, q = 3 and r = -7 set by default against the block titled “**Find the roots of.”**

Enter the quadratic equation of variable x with different values of p, q, and r for which the solution is required. The user can also incorporate **higher-order equations** of degrees greater than two depending on the requirement.

### Step 2:

Click the **Submit** button after entering the polynomial. The calculator computes the roots of the function by putting it equal to zero.

### Output:

The **calculator** processes the input equation which opens the following output windows.

### Input Interpretation:

The calculator interprets the input polynomial and displays the equation for the user for which the roots are to be determined.

### Results:

This window shows the roots or solutions for the equation. These are the x-intercepts with y = 0. These roots can be **real** or **imaginary** depending upon the **discriminant** value in the quadratic formula.

The **quadratic formula** for the quadratic equation:

\[ (p)x^2 + (q)x + r = 0 \]

is

\[ x = \frac{ -q \pm \sqrt{ q^2 – 4pr } } { 2p } \]

Here, the value of discriminant:

\[ D = q^2 – 4(p)(r) \]

determines the roots to be real or imaginary.

### Root Plot:

The root plot shows the graph in the 2D plane for the input equation. The **roots** are represented by **dots on the x-axis**. The imaginary roots are displayed in the complex plane.

### Number Line:

This window displays the roots of the equation on the number line.

### Sum of Roots:

This window is displayed when there are numerous roots. The **roots are added** and their sum is obtained.

### Product of Roots:

This window displays the product of all the roots by **multiplying** them simultaneously.

## Solved Examples

Here are some examples that can be solved using the Root Finder calculator.

### Example 1

Find the roots for the equation:

\[ x^2 + 4x – 7 \]

### Solution

Using the equation:

\[ x^2 + 4x – 7 = 0 \]

Input the above-mentioned equation in the calculator.

The quadratic formula is used to find the roots of the quadratic equation:

\[ (p)x^2 + (q)x + r = 0 \]

The formula is given as:

\[ x = \frac{ -q \pm \sqrt{ q^2 – 4pr } } { 2p } \]

Step-wise solution of the problem is given as:

Here,

**p = 1**

**q = 4**

**r = -7**

\[ x = \frac{ -4 \pm \sqrt{ (4)^2 – 4(1)(-7) } } { 2(1) } \]

\[ x = \frac{ -4 \pm \sqrt{ 16 + 28 } } { 2 } \]

\[ x = \frac{ -4 \pm \sqrt{ 44 } } { 2 } \]

\[ x = \frac{ -4 \pm 2\sqrt{ 11 } } { 2 } \]

\[ x = -2 \pm \sqrt{ 11 } \]

So the** roots** are

\[ x = -2 + \sqrt{ 11 } , -2 – \sqrt{11} \]

Figure 1 shows the roots of example 1.

The sum of roots S is;

\[ S = (-2 + \sqrt{ 11 }) + (-2 – \sqrt{11}) \]

\[ S = (-2 -2) + ( \sqrt{ 11 } – \sqrt{11}) = -4 + 0 = -4 \]

And the product of roots P is:

\[ P = ( -2 + \sqrt{ 11 } )( -2 – \sqrt{11} ) \]

\[ P = 4 + 2\sqrt{ 11 } -2)\sqrt{ 11 } – 11 = 4 + 0 – 11 = -7 \]

The same results are obtained using the calculator.

### Example 2

Find the roots for the equation:

\[ x^2 – 6x + 9 \]

### Solution

Put the given equation in the calculator:

\[ x^2 – 6x + 9 = 0 \]

Quadratic formula is given as:

\[ x = \frac{ -q \pm \sqrt{ q^2 – 4pr } } { 2p } \]

Given that:

**p = 1**

**q = -6**

**r = 9**

Step-wise Solution is given below.

The formula becomes:

\[ x = \frac{ -(-6) \pm \sqrt{ (-6)^2 – 4(1)(9) } } { 2(1) } \]

\[ x = \frac{ 6 \pm \sqrt{ 36 – 36 } } { 2 } \]

\[ x = \frac{ 6 \pm \sqrt{ 0 } } { 2 } \]

\[ x = \frac{ 6 \pm 0 } { 2 } \]

\[ x = \frac{ 6 } { 2 } \]

**x = 3**

So the **root** of the above equation is 3.

Figure 2 shows the root of example 2.

The same results are obtained using the calculator.

### Example 3

Find the roots for the equation given below:

\[x^3 + 2x^2 – 5x -10\]

### Solution

Input the following equation in the calculator to obtain the roots:

\[ x^3 + 2x^2 – 5x -10 = 0 \]

Step-wise Solution is given as:

Using the factorization method:

Take ( x + 2 ) as a common factor.

\[ x^2 ( x + 2 ) – 5 ( x +2 ) = 0\]

\[( x + 2 ) ( x^2 – 5 ) = 0\]

**( x + 2 ) = 0**

**x = -2**

\[ ( (x)^2 – 5 ) = 0\]

\[(x)^2 = 5\]

\[ \sqrt{x^2} = \sqrt{5}\]

\[ x = \pm \sqrt{5}\]

So the **roots** are

**x = -2**

\[\sqrt{5} \]

\[-\sqrt{5} \]

Figure 3 shows the roots of example 3.

The sum of roots S is:

\[ S= -2 + \sqrt{5} + (-\sqrt{5}) = -2 + 0 = -2 \]

The product of roots P is:

\[ P = (-2) (\sqrt{5}) (-\sqrt{5}) = 2(5) = 10 \]

The same results are obtained using the calculator.** **

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*All the images are created using GeoGebra. *