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Algebra Tutoring

How to Solve a Quadratic Equation?

Before looking at the ways of solving a quadratic equation let’s learn what does it mean by a quadratic equation. Given below is a simple quadratic equation:

ax² + bx + c = 0 , where ‘x’ is a variable and ‘a’,’b’ & ‘c’ are constant numbers and (a ≠ 0). If (a = 0) then this quadratic equation will turn into a linear equation.
A quadratic equation is a polynomial equation of a single variable(univariate) of the second degree. In the above equation ‘x’ is the univariate and ‘a’, ‘b’, ‘c’ are the constants known as quadratic coefficient, linear coefficient and the constant term respectively.

Let’s now try to solve the above equation. There are five general ways of solving a quadratic equation:

  1. Factoring
  2. Taking roots
  3. Completing the square
  4. Quadratic formula
  5. Graphing

We will see the methods one by one.

  1. Factoring– Here first you have to factor the quadratic equation. We need 2 factors of ‘c’ which adds up to be equal to ‘b’. Let’s take an actual example here

x² + 7x + 12 = 0

Here we need 2 factors of 12 which sums up to 7, i.e. 4 & 3. So now,

(x+3)(x+4) = 0  As the product of these two components are zero hence we can conclude that either of them is equal to zero for the product to be zero which means either

X+3 = 0  or   x+4 = 0 , hence x = -3 or x = -4. So the soultion is X = -3,-4

  1. Taking rootsThere are different types of quadratic equation. Few are of a type where you can equate it on both sides and take the square root of both sides to find the value of ‘x’. Eg.:

x² – 16 = 0            =>  x² = 16          =>  √x² = √16    =>  x = ±4

  1. Completing the Squarefor those equations which can’t be factored neither can be taken roots of we need to change it to a form where we complete the square to then square root it to find the value. Eg:

x² – 4x – 8 = 0      =>  x² – 4x = 8     => x² – 4x + 4 = 8 + 4 (+4 added to both sides)    => (x – 2)² = 12 => √(x – 2)² = √12

x – 2 = ±√12       =>  x = 2 ± √12   => x = 2 ± 2√3

  1. Quadratic FormulaThis method was derived by someone during seventh century in India while solving a quadratic equation by completing the square method. He found the steps in the solution can be put exactly in a formula i.e.

to solve an equation that is ax² + bx + c = 0 .

  1. Graphing method – This is not the most preferred or practised method and also has lot of difficulties in it. The results here are taken from a graph based on the fact that ax² + bx + c = 0 can be considered the x-intercepts of y= ax² + bx + c = 0 and hence a parabolic graph is formed. There is a risk of nonaccurate results here.