We will discuss that a quadratic equation has only two roots or in other words we can say that a quadratic equation cannot have more than two roots.
We will prove this one-by-one.
A quadratic equation has only two roots.
Proof:
Let us, consider the quadratic equation of the general form
ax\(^{2}\) + bx + c = 0, (a ≠ 0) ............... (i)
Now divide each term by a (since, a ≠ 0), we get
x\(^{2}\) + \(\frac{b}{a}\)x + \(\frac{c}{a}\) = 0
⇒ x\(^{2}\) + 2 * x * \(\frac{b}{2a}\) + (\(\frac{b}{2a}\))\(^{2}\) – (\(\frac{b}{2a}\))\(^{2}\) + \(\frac{c}{a}\) = 0
⇒ (x + \(\frac{b}{2a}\))\(^{2}\) - \(\frac{b^{2} - 4ac}{4a^{2}}\) = 0
⇒ (x + \(\frac{b}{2a}\))\(^{2}\) – \((\frac{\sqrt{b^{2} - 4ac}}{2a})^{2}\) = 0
⇒ (x + \(\frac{b}{2a}\) + \(\frac{\sqrt{b^{2} - 4ac}}{2a}\))(x + \(\frac{b}{2a}\) - \(\frac{\sqrt{b^{2} - 4ac}}{2a}\)) = 0
⇒ [x - \((\frac{-b - \sqrt{b^{2} - 4ac}}{2a})\)][x - \((\frac{-b + \sqrt{b^{2} - 4ac}}{2a})\)] = 0
⇒ (x - α)(x - β) = 0, where α = \(\frac{- b - \sqrt{b^{2} - 4ac}}{2a}\) and β = \(\frac{- b + \sqrt{b^{2} - 4ac}}{2a}\)
Now we can clearly see that the equation ax\(^{2}\) + bx + c = 0 reduces to (x - α)(x - β) = 0 and the equation ax\(^{2}\) + bx + c = 0 is only satisfied by the values x = α and x = β.
Except α and β no other values of x satisfies the equation ax\(^{2}\) + bx + c = 0.
Hence, we can say that the equation ax\(^{2}\) + bx + c = 0 has two and only two roots.
Therefore, a quadratic equation has two and only two roots.
Solved example on quadratic equation:
Solve the quadratic equation x\(^{2}\) - 4x + 13 = 0
Solution:
The given quadratic equation is x\(^{2}\) - 4x + 13 = 0
Comparing the given equation with the general form of the quadratic equation ax\(^{2}\) + bx + c = 0, we get
a = 1, b = -4 and c = 13
Therefore, x = \(\frac{- b ± \sqrt{b^{2} - 4ac}}{2a}\)
⇒ x = \(\frac{- (-4) ± \sqrt{(-4)^{2} - 4(1)(13)}}{2(1)}\)
⇒ x = \(\frac{4 ± \sqrt{16 - 52}}{2}\)
⇒ x = \(\frac{4 ± \sqrt{-36}}{2}\)
⇒ x = \(\frac{4 ± 6i}{2}\), [Since i = √-1]
⇒ x = 2 ± 3i
Hence, the given quadratic equation has two and only two roots.
The roots are 2 + 3i and 2 - 3i.
11 and 12 Grade Math
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