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Common Roots

In this section we are going to discuss about conditions when two quadratic expressions or equations have common roots. Both roots may be coomon or just one root.

Both roots are common

If two quadratic equations \(a_1x^2 + b_1x + c_1 = 0\) and \(a_2x^2 + b_2x + c_2 = 0\) have both roots in common, then the ratios of their corresponding coefficients are equal, i.e., \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

Proof

Let the common roots of the quadratic equations \(a_1x^2 + b_1x + c_1 = 0\) and \(a_2x^2 + b_2x + c_2 = 0\) be \(\alpha\) and \(\beta\).

  1. Sum of the Roots:

  2. For the first equation, the sum of the roots, \(\alpha + \beta\), is \(-\frac{b_1}{a_1}\).

  3. For the second equation, the sum of the roots is \(-\frac{b_2}{a_2}\).

  4. Since \(\alpha\) and \(\beta\) are the roots of both equations, these sums must be equal: \( -\frac{b_1}{a_1} = -\frac{b_2}{a_2} \)

    Simplifying, we get: \( \frac{b_1}{b_2} = \frac{a_1}{a_2} \) (Equation 1)

  5. Product of the Roots:

  6. For the first equation, the product of the roots, \(\alpha \beta\), is \(\frac{c_1}{a_1}\).

  7. For the second equation, the product of the roots is \(\frac{c_2}{a_2}\).

  8. As \(\alpha\) and \(\beta\) are the roots of both equations, these products must be equal: \( \frac{c_1}{a_1} = \frac{c_2}{a_2} \)

    Simplifying, we get: \( \frac{c_1}{c_2} = \frac{a_1}{a_2} \) (Equation 2)

Combining Equation 1 and Equation 2, we have:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

This completes the proof. Hence, for two quadratic equations to have both roots in common, the ratios of their corresponding coefficients must be equal.

Exactly one root is common

For two quadratic equations \( a_1x^2 + b_1x + c_1 = 0 \) and \( a_2x^2 + b_2x + c_2 = 0 \), if they have exactly one root in common, the following condition holds:

\[ \left( \det \begin{pmatrix} a_1 & a_2 \\ c_1 & c_2 \end{pmatrix} \right)^2 = \left( \det \begin{pmatrix} a_1 & a_2 \\ b_1 & b_2 \end{pmatrix} \right) \left( \det \begin{pmatrix} b_1 & b_2 \\ c_1 & c_2 \end{pmatrix} \right) \]

Where \(\det\) represents the determinant of a matrix.

Proof

Let \(\alpha\) be the common root of the two quadratic equations:

  1. \( a_1x^2 + b_1x + c_1 = 0 \)
  2. \( a_2x^2 + b_2x + c_2 = 0 \)

Substitute \(\alpha\) into both equations:

  1. \( a_1\alpha^2 + b_1\alpha + c_1 = 0 \)
  2. \( a_2\alpha^2 + b_2\alpha + c_2 = 0 \)

Apply the Cross-Multiplication Method to write:

\[ \frac{\alpha^2}{(b_1c_2 - b_2c_1)} = \frac{\alpha}{(c_1a_2 - c_2a_1)} = \frac{1}{(a_1b_2 - a_2b_1)} \]

We now eliminate \(\alpha\) by equating the first and last parts of the proportion:

\[ \frac{\alpha^2}{(b_1c_2 - b_2c_1)} = \frac{1}{(a_1b_2 - a_2b_1)} \]

By cross-multiplying to solve for \(\alpha\), we obtain:

\[ \alpha^2(a_1b_2 - a_2b_1) = (b_1c_2 - b_2c_1) \]

Now, since \(\alpha\) is not zero, we can square the middle part of the proportion and set it equal to the product of the other two parts to eliminate \(\alpha\):

\[ \left(\frac{\alpha}{(c_1a_2 - c_2a_1)}\right)^2 = \frac{\alpha^2}{(b_1c_2 - b_2c_1)} \cdot \frac{1}{(a_1b_2 - a_2b_1)} \]

This yields the condition for the coefficients when one root is common:

\[ (c_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1) \]

This completes the proof. We can write this result in determinant form.

Rational coefficients and irrational roots

Let there be two quadratic equations with rational coefficients:

  1. \( a_1x^2 + b_1x + c_1 = 0 \) with roots \(\alpha\) and \(\beta\). It is known that discriminant of this equation is not a perfect square.
  2. \( a_2x^2 + b_2x + c_2 = 0 \) sharing the common root \(\alpha\).

Since the discriminant of the first equation is not a perfect square, the roots \(\alpha\) and \(\beta\) are not rational numbers, but they are conjugate surds. Conjugate surds are expressions of the form \(a + \sqrt{b}\) and \(a - \sqrt{b}\), where \(a\) and \(b\) are rational numbers, and \(b\) is not a perfect square.

Now, for the second equation, which shares the root \(\alpha\), the other root must also be \(\beta\) because the coefficients are rational, and by the Rational Root Theorem, if one irrational root is present, its conjugate must also be a root to ensure that all the coefficients remain rational.

Therefore, both quadratic equations have the same pair of irrational roots, \(\alpha\) and \(\beta\), which means that the equations are proportional to each other. The final condition, which establishes the relationship between the coefficients of these two proportional quadratic equations, is:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

This condition implies that the two quadratic equations are essentially the same equation, just multiplied by different non-zero constants.