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Theory:

The solution of the quadratic equation is the value of the variable that makes the equation zero. We can say these solutions as roots or zeroes.
There are three methods to solve the quadratic equations.
  • Factorisation
  • Completing the square
  • Quadratic formula
Solve by factorisation method
Let us see the procedure to solve the quadratic equation by the factorisation method.
Step 1: Write the given equation in standard form.
 
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
 
Step 3: Group the expression into two linear factors by taking the common term outside.
 
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Important!
Generally, we say \(\alpha\) is a root of the quadratic equation \(ax^2 + bx + c = 0\) if \(a \alpha^2 + b \alpha + c = 0\). So that means \(x = \alpha\) is a solution of the quadratic equation.
Example:
Find the roots of the quadratic equation \(2x^2 + 4 = 9x\).
 
Solution:
 
The given equation is \(2x^2 + 4 = 9x\).
 
Let us first write the given equation in standard form.
 
\(2x^2 - 9x + 4 = 0\)
 
Now, split the middle term by the above procedure.
 
\(2x^2 - x - 8x + 4 = 0\)
 
Group the expression into two linear factors by taking the common term outside.
 
\(x(2x - 1) - 4(2x - 1) = 0\)
 
\((2x - 1)(x - 4) = 0\)
 
Now, solve for \(x\) by equating each linear factor to zero.
 
\(2x - 1 = 0\) or \(x - 4 = 0\)
 
\(x =\) 12 or \(x = 4\)
 
Therefore, the roots of \(2x^2 + 4 = 9x\) are 12 and \(4\).
Important!
Any quadratic equation can have at most two roots.