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
From Factor Theory of Quadratic Equation to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.