Q.17. Method of solving Quadratic Equations

Question 17.
Please explain the proper method on "Solving Quadratic Equations by Extraction of Roots"?

Answer 17.
A quadratic equation is of the form
ax^2 + bx + c = 0, where a, b, and c are real constants and a is not zero.
If a = 0, then it is no longer a quadratic equation.
It, then, becomes a linear equation as the term ax^2 will then be zero.
The equation can be written as
a [ x^2 + (b/a) x + (c/a) ] = 0 [because a ≠ 0]

Now, using first two terms, make a perfect square.
To do that we have to add and subtract (b/2a)^2. Thus, equation is
a [ x^2 + (b/a) x + (b/2a)^2 - (b/2a)^2 + (c/a) ] = 0
=> a [ { x + (b/2a) }^2 - (b^2 - 4ac) / 4a^2 ] = 0
=> [ x + (b/2a) ]^2 = (b^2 - 4ac) / 4a^2 [ because a is not zero ]
=> x + (b/2a) = +/- sqrt (b^2 - 4ac) / 2a
=> x = - (b/2a) +/- sqrt (b^2 - 4ac) / 2a
These are the roots of the quadratic equation.

Now let us take an example.
If the given quadratic equation is 2x^2 + 5x +2 = 0
Comparing it with ax^2 + bx + c = 0, a = 2, b = 5 and c = 2.
Putting them in the formula of roots derived above,
x = - (5/2*2) +/- sqrt (5^2 - 4*2*2) / 2*2
=> x = - 5/4 +/- 3/4 = - 1/2 or - 2

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