It is important for you to check how to solve the quadratic equation in the real and complex number system to accurately understand how the roots are determined from the initial equation. That way, you will have a background understanding of why the roots are very important for determining the main equation in a reverse computation.
The Quadratic Equation Roots Formula
The roots of a quadratic are the solutions of x that are gotten from the standard quadratic equation given as ax2 + bx + c = 0
To find the roots of a quadratic equation, it is important to first understand what a quadratic equation is. The Quadratic Equation is a type of equation whose standard form is
ax2 + bx + c = 0 …….. Eqn (1)
where a, b & c are all real numbers.
The main formula for determining the quadratic equation is through the quadratic formula, which is given as
Apart from the General quadratic formula, Factoring and Completing the Squares are two other methods for getting the solutions to the quadratic equation.
This article focuses on the factorization method and the general equation that allows easy formation of the quadratic equation when its roots are known.
Recall that the roots of the equation mean the solution of the same solution. So to use them to find the quadratic equation means calculating from reverse until the standard equation form is reached. It is extremely important to know how to find the roots of the quadratic equation for this method to make sense.
How to Find a Quadratic Equation Using its Roots?
Method 1: Defactorization method
Step 1: let x be made equal to the two solutions
Step 2: Move the solutions of x to the left-hand side.
Step 3: Multiply the two equations together
Step 4: expand the equation at the left-hand side of the zero to form the standard quadratic equation
Method 2: Using the Basic formula
Step 1: write out the equation x2 – (sum of the roots) X x + (products of the roots)) = 0
Step 2: Find the sum and product of the Roots and input them into the formula
How to Find a Quadratic Equation Using its Roots?
Before determining the root of the quadratic equation we will first show how the factorization method works for finding
Consider the quadratic equation x – 2x – 8 = 0 and find its roots using the factorization method
Solution
X2 – 2x – 8 = 0
X2 + 2x – 4x – 8 = 0
(x-4)(x + 2) = 0
x + 2 = 0, x – 4 = 0
x = – 2, x = 4
These are the roots of the equation
Now we will use the roots gotten from the equation x – 2x – 8 = 0 to find the equation in reverse
Example 1:
Find the standard equation of these roots x = 2, 4
Solution
Step 1: let x be made equal to the two solutions
X = – 2, x = 4
Step 2: Move the solutions of x to the left-hand side.
This will make x = – 2 become x + 2 = 0
And x = 4 will become x – 4 = 0
So we have x + 2 = 0, x – 4 = 0
Step 3: Multiply the two equations together
This means that multiply x – 4 = 0 by x + 2 = 0
Clearly, since both equations is actually equal to zeros, then their multiplication will no doubt be zero, so the multiplication is correct as 0 = 0
(x-4)(x + 2) = 0
Step 4: expand the equation at the left-hand side of the zero to form the standard quadratic equation
(x-4)(x + 2) = 0
X2 + 2x – 4x – 8 = 0
X2 – 2x – 8 = 0
It is clear that the above-derived equation is the same as the example we gave above. Clearly, it is the equation of the roots.
X2 – 2x – 8 = 0
X2 + 2x – 4x – 8 = 0
(x-4)(x + 2) = 0
x + 2 = 0, x – 4 = 0
x = – 2, x = 4
These are the roots of the equation
Now we will also use the second method, which is considered the easier method of the two.
Example 2
Find the quadratic equation of the two roots x = -2, 4
Solution.
We will use the second method which is given as
x2 – (sum of the roots) X x + (products of the roots) = 0
Step 2: find the sum and product of the Roots and input into the formula
The sum of the two roots are – 2 and 4 = 2. The products of the two roots i.e – 2 x 4 = – 8
Now put them into the equation
x2 – (2) X x + (-8)) = 0
x2 – 2 X x – 8 = 0
Applying BODMAS, the two middles must be multiplied together before any other computation.
x2 – 2x – 8 = 0
As can be seen from the above computations, the roots gave the same answer as the first. This, therefore, shows that this computation is just as correct.
It is important to understand that the second method is more advisable for performing computations as it can work for all types of roots, a major advantage over the defactorization method.
We will consider another example to prove this.
Example 3
Find the Quadratic equation of the two roots 2 + √3 and 2 – √3.
Solution
Now we will go with the second method
x2 – (sum of the roots) X x + (products of the roots)) = 0
The product of the two roots is
(2 + √3)(2 – √3) = 4 – 2√3 +2√3 – √3 X √3
= 4 – 0 – 3
= 1
The sun if the two roots is
2 + √3 + 2 – √3 = 2 + 2 + √3 – √3
= 4
Putting the two into the formula will give
x2 – (4) X x + (1) = 0
x2 – 4x + 1 = 0
Trying to use the defactorization method for this solution will be impossible.
Conclusion
Knowing how to find a quadratic equation through its roots is extremely important. The roots of the quadratic equation are well connected to the complex number, and understanding its calculation and workings prepare students for the imaginary number series. This article has considered several examples to help students have a better understanding of how to derive the equation of any root.
