DERIVATION OF QUADRATIC FORMULA - LUCID EXPLANATION, EXAMPLE, EXERCISE

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Please study
about Quadratic Equation before Derivation Of Quadratic Formula,
if you have not already done so.

There, we discussed about Quadratic Polynomial
or Quadratic Expression, Quadratic Equation,
Methods to Solve a Quadratic Equation and the first
method of Solving Quadratic Equations by Factoring.

Here we discuss the second method.

We derive the Formula used.

An Example in applying the formula is given.

Another problem for practice is given.

Derivation of Quadratic formula :

To solve ax2 + bx + c = 0 where a ( ≠ 0 ), b, c
are constants which can take real number values.

ax2 + bx + c = 0 ⇒ ax2 + bx = -c
Dividing by a on both sides, we get
x2 + (ba)x = -ca
x2 + 2x(b⁄2a) = -ca .........(i)

The L.H.S. of equation(i) has (first term)2
and 2(first term)(second term) terms
where fist term = x and second term = (b⁄2a).

If we add (second term)2 { = (b⁄2a)2 },
the L.H.S. of equation(i) becomes a perfect square.

Adding (b⁄2a)2 to both sides of equation(i), we get
x2 + 2x(b⁄2a) + (b⁄2a)2 = -ca + (b⁄2a)2
⇒ (x + b⁄2a)2 = b2⁄4a2 - ca = ( b2 - 4ac)⁄(4a2)
⇒ (x + b⁄2a) = ±√{( b2 - 4ac)⁄(4a2)} = ±√( b2 - 4ac)⁄2a
x = -b⁄2a ± √(b2 - 4ac)⁄2a
x = {-b ± √(b2 - 4ac)}⁄2a
This is the Quadratic Formula.

ax2 + bx + c = 0 ⇒ x = {-b ± √(b2 - 4ac)}⁄2a

Thus, the two roots of ax2 + bx + c = 0 are
{-b + √(b2 - 4ac)}⁄2a and {-b - √(b2 - 4ac)}⁄2a.

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Example : Derivation of Quadratic Formula

Solve the following equation using Quadratic Formula

x2 + x - 42 = 0.

Solution of Example of Derivation of Quadratic Formula :

To solve x2 + x - 42 = 0 using Quadratic Formula
Comparing this equation with ax2 + bx + c = 0, we get
a = 1, b = 1 and c = -42
We know by Quadratic Formula, x = {-b ± √(b2 - 4ac)}⁄2a
Applying this Quadratic Formula here, we get
x = {-b ± √(b2 - 4ac)}⁄2a
= [ (-1) ± √{(1)2 - 4(1)(-42)}]⁄2(1)
= [ (-1) ± √{1 + 168}]⁄2(1)
= [ (-1) ± √{169}]⁄2(1) = [ (-1) ± 13]⁄2(1)
= (-1 + 13)⁄2, (-1 - 13)⁄2 = 12⁄2, -14⁄2
= 6, -7 Ans.

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Exercise : Derivation of Quadratic Formula

Problem on Derivation of Quadratic Formula :

Solve the following equation using Quadratic Formula
x2 - 4x - 12 = 0.

The following Links take you to
further study of Quadratic Formula.

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Set 1 of Problems on Quadratic Formula.

Set 2 of Problems on Quadratic Formula.

Set 1 of Problems on Sum and Product of Roots.

Set 2 of Problems on Sum and Product of Roots.

Finding Quadratic Equation whose roots are given.

Nature of the roots of a Quadratic Equation.

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Answer to Exercise : Derivation of Quadratic Formula

Answer to problems on Derivation of Quadratic Formula :

6, -2

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