Quadratic Equation has Only Two Roots

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 

From Quadratic Equation has Only Two Roots to HOME PAGE