We will discuss about the factor theory of quadratic equation.
Suppose when we assume that β be a root of the quadratic equation ax\(^{2}\) + bx + c = 0, then we get (x  β) is a factor of the quadratic expression ax^2 + bx + c.
Conversely, when we assume that (x  β) is a factor of the quadratic expression ax\(^{2}\) + bx + c then β is a root of the quadratic equation ax\(^{2}\) + bx + c = 0.
Proof:
The given quadratic equation ax\(^{2}\) + bx + c = 0
According to the problem, β is a root of the quadratic equation
ax\(^{2}\) + bx + c = 0
Hence, aβ\(^{2}\) + bβ + c = 0
Now, ax\(^{2}\) + bx + c
= ax\(^{2}\) + bx + c  (aβ\(^{2}\) + bβ + c), [Since, aβ\(^{2}\) + bβ + c = 0]
= a(x\(^{2}\)  β\(^{2}\)) + b(x  β)
= (x  β)[a(x + β) + b]
Therefore, we clearly see that (x  β) is a factor of the quadratic expression ax\(^{2}\) + bx + c.
Conversely, when (x  α) is a factor of the quadratic expression ax\(^{2}\) + bx + c then we can express,
ax\(^{2}\) + bx + c = (x  α)(mx + n), where m (≠ 0) and n are constants.
Now, we need to substitute x = β on both sides of the identity ax\(^{2}\) + bx + c = (x  β)(mx + n) then we get,
aβ\(^{2}\) + bβ + c = (β  β) × (mβ + n) = 0
aβ\(^{2}\) + bβ + c = 0 × (mβ + n) = 0
It is evident that the equation ax\(^{2}\) + bx + c = 0 is satisfied by x = β.
Therefore, β is a root of the equation ax\(^{2}\) + bx + c = 0.
`11 and 12 Grade Math
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