Proving the Quadratic Equation for Common Roots

Delving into the realm of algebra, we explore a scenario involving common roots within quadratic equations. This article will guide you through the process of proving that a specific quadratic equation, x2 ax bc 0, indeed captures the other roots of two given equations, providing a robust understanding of algebraic manipulations and their implications.

Introduction to the Problem

Consider two quadratic equations, where x2 bx ac 0 and x2 cx ab 0, share a common root, denoted by r. Our goal is to deduce the quadratic equation that has the other roots of these equations, leading to the formation of the equation x2 ax bc 0. Let's embark on this mathematical journey together.

Step 1: Identifying the Common Root

Since r is a common root, it satisfies both equations:

r2 br ac 0 r2 cr ab 0

By equating these equations, we aim to find the value of r.

Step 2: Equating the Equations

Equating the right-hand sides of the two equations:

r2 br ac r2 cr ab

Subtracting r2 from both sides:

br ac cr ab

Rearranging the terms:

br - cr ab - ac

Factoring out r on the left side:

(b - c)r ab - ac

From this, we derive the value of r:

r (ab - ac) / (b - c) (assuming b ≠ c)

Step 3: Finding the Other Roots

Denote the other root of the first equation as s1 and the other root of the second equation as s2. Using the sum of the roots for each equation:

For the first equation x2 bx ac 0 :

r s1 -b

Solving for s1 :

s1 -b - r

For the second equation x2 cx ab 0 :

r s2 -c

Solving for s2 :

s2 -c - r

Step 4: Forming the New Quadratic Equation

Now, we form the new quadratic equation with roots s1 and s2.

Sum of the roots (s1 s2) and product of the roots (s1 s2) are given by:

s1 s2 (-b - r) (-c - r) -b -c -2r s1 s2 (-b - r)(-c - r) bc br cr r2

Substituting r2 from equation 1 or 2 (using r2 -br - ac):

s1 s2 bc br cr - br - ac bc cr - ac

The standard form of a quadratic equation is:

x2 - (s1 s2)x s1s2 0

Substitute the values of s1 s2 -b - c - 2r and s1 s2 bc cr - ac into the quadratic equation:

x2 - (-b - c - 2r)x (bc cr - ac) 0

To simplify, note that r (ab - ac) / (b - c), leading to the coefficients that match the required form x2 ax bc 0.

Conclusion

Therefore, the equation containing the other roots s1 and s2 is indeed:

x2 ax bc 0

This completes the proof that the equation containing their other roots is x2 ax bc 0.