Vidhyamath Forum

Quadratic Equation

Rita Lamichhane

Student

Prove that when the quadratic equations $�^{2} + � � + � � = 0$ and $�^{2} + � � + � � = 0$ share a common root, their remaining roots will satisfy the equation $�^{2} + � � + � � = 0$ .

29 Dec, 2023

Replies (1)

Harit Thakuri

Instructor

Let's analyze the given equations and deductions:

The given quadratic equations are:
$�^{2} + � � + � � = 0$ ----(1)
$�^{2} + � � + � � = 0$ ----(2)
Let the roots of equation (1) be $�$ and $�$ , and the roots of equation (2) be $�$ and $�$ .
Since $�$ is a common root, $�^{2} + � � + � � = �^{2} + � � + � �$ .
Simplifying, $(� - �) � = � � - � �$ , which implies $� = �$ .
Now, $� + � = - �$ and $� + � = - �$ .

Since $� = �$ , we can rewrite the above equations as $� + � = - �$ and $� + � = - �$ .
From these equations, $� = �$ and $� = �$ .
Since $�$ is a common root, $�^{2} + � � + � � = 0$ .
Substituting $� = �$ , we get $�^{2} + � � + � � = 0$ .

Factoring out $�$ , we get $� (� + � + �) = 0$ , which implies $� + � + � = 0$ or $� + � = - �$ ----(4).
Now, let's find the equation having roots $�$ and $�$ :
$�^{2} - (� + �) � + � � = 0$ .
Substituting $� = �$ and $� = �$ , we get $�^{2} + � � + � � = 0$ .
Therefore, the equation having roots $�$ and $�$ is $�^{2} + � � + � � = 0$ which is the required equation.

:) Happy Learning

30 Dec, 2023

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Quadratic Equation

Rita Lamichhane

Replies (1)

Harit Thakuri

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