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Nature of Roots

Number of roots

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) provides crucial information about the nature of its roots. The discriminant, denoted as \(\Delta\), is calculated using the formula:

\[\Delta= b^2 - 4ac \]

Depending on the value of \(\Delta\), we can determine the nature of the roots as follows:

Discriminant Positive (\(\Delta> 0 \))

  • Nature of Roots: Two distinct real roots.
  • Explanation: A positive discriminant implies that the square root of \(\Delta\) is a real number. Since the quadratic formula includes a \( \pm \) sign, it means there will be two different values for \( x \), hence two distinct real roots \(\alpha\) and \(\beta\).
  • Roots: \( \alpha = \frac{-b + \sqrt{\Delta}}{2a} \) and \( \beta = \frac{-b - \sqrt{\Delta}}{2a} \).

Discriminant Zero (\(\Delta= 0 \))

  • Nature of Roots: One real root, also known as a repeated or double root.
  • Explanation: A discriminant of zero means that the square root of \(\Delta\) is zero. Thus, the quadratic formula reduces to a single value for \( x \), indicating that both roots are the same.
  • Roots: \( \alpha = \beta = \frac{-b}{2a} \).

Discriminant Negative (\(\Delta< 0 \))

  • Nature of Roots: Two complex conjugate roots. (If you do not understand complex numbes, then for now you can just assume that roots are not real.)
  • Explanation: A negative discriminant means that the square root of \(\Delta\) involves the square root of a negative number, which is not real. In this case, the roots are complex numbers and are conjugates of each other (they have the same real part but opposite imaginary parts).
  • Roots: \( \alpha = \frac{-b}{2a} + i\frac{\sqrt{-D}}{2a} \) and \( \beta = \frac{-b}{2a} - i\frac{\sqrt{-D}}{2a} \), where \( i \) is the imaginary unit.

Rational Roots

Theorem

For a quadratic equation \( ax^2 + bx + c = 0 \) with rational coefficients \( a \), \( b \), and \( c \), the roots are rational if and only if the discriminant \( D \), where \( D = b^2 - 4ac \), is a perfect square of a rational number.

Proof

The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

For the roots to be rational, the entire expression under the quadratic formula must be rational. This requirement hinges primarily on the term \( \sqrt{D} \), as \( -b \) and \( 2a \) are rational (since \( a \), \( b \), and \( c \) are rational).

The square root of a number is rational if and only if that number is a perfect square within the rational numbers. Therefore, \( \sqrt{D} \) is rational if and only if \( D \) is a perfect square of a rational number. If this is the case, then both \( \frac{-b + \sqrt{D}}{2a} \) and \( \frac{-b - \sqrt{D}}{2a} \) will be rational, and consequently, the roots of the quadratic equation will be rational.

On the other hand, if \( D \) is not a perfect square in the rational numbers, then \( \sqrt{D} \) is irrational. This would make at least one of the roots irrational since the addition or subtraction of an irrational number to or from a rational number results in an irrational number.

Thus, the roots of a quadratic equation \( ax^2 + bx + c = 0 \) with rational coefficients are rational if and only if the discriminant \( D \) is a perfect square of a rational number. This condition is pivotal for the rationality of the roots, providing a clear criterion for determining the nature of the solutions of a quadratic equation with rational coefficients.

Irrational Roots

Theorem

Given a quadratic equation with rational coefficients:

\[ ax^2 + bx + c = 0 \]

where \(a\), \(b\), and \(c\) are rational numbers and \(a \neq 0\), if one root of the equation is irrational, then the other root must also be irrational, and furthermore, the two roots are conjugate surds. This occurs when the discriminant \(b^2 - 4ac\) is not a perfect square of a rational number.

Proof

Let there be a quadratic equation with rational coefficients:

\[ ax^2 + bx + c = 0 \]

with \(a\), \(b\), and \(c\) being rational numbers.

The roots of the quadratic equation are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The discriminant of this equation is \(\Delta = b^2 - 4ac\).

  1. Case 1: Discriminant is a Perfect Square(We already considered this above)

    If \(\Delta\) is a perfect square, then \(\sqrt{\Delta}\) is a rational number. Therefore, both \( \frac{-b + \sqrt{\Delta}}{2a} \) and \( \frac{-b - \sqrt{\Delta}}{2a} \) are rational, which means that both roots of the equation are rational.

  2. Case 2: Discriminant is Not a Perfect Square

    If \(\Delta\) is not a perfect square, then \(\sqrt{\Delta}\) is irrational. This implies that \( \frac{-b + \sqrt{\Delta}}{2a} \) and \( \frac{-b - \sqrt{\Delta}}{2a} \) are both irrational because the sum or difference of a rational number and an irrational number is irrational.

Clearly, if one root is irrational of the form \(r + s\sqrt{\Delta}\), the other root must be its conjugate \(r - s\sqrt{\Delta}\). In other words, roots are conjugate surds.

Thus, if one root of a quadratic equation with rational coefficients is irrational, the other root is also irrational, and both roots are conjugate surds, which occurs precisely when the discriminant is not a perfect square of a rational number.

Integer Roots

Theorem: In a monic quadratic expression \( x^2 + bx + c \) with integer coefficients, the roots are integers if and only if the discriminant \( \Delta = b^2 - 4c \) is a perfect square of an integer.

Proof

The roots of the quadratic expression are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{\Delta}}{2} \]

where \( \Delta = b^2 - 4c \).

đź’ˇ The main idea of the proof is that roots can be integers only when \(\Delta\) under the square root is a perfect square of an integer, because if it is not so, then roots will be irrational. There is a problem though. We have 2 in the Denominator. We show that this does not matter because the expression in the Numerator is always even. A careful consideration leads us to the conclusion that \(c\) is insignificant and its \(b\) which needs a detailed analysis.

Analysis Based on the Nature of \( b \):

  1. When \( b \) is an Even Integer:

    • In this case, \( b \) can be written as \( b = 2k \) for some integer \( k \).
    • The discriminant \( \Delta \) becomes \( \Delta = (2k)^2 - 4c = 4k^2 - 4c \).
    • Notice that \( \Delta \) is divisible by 4, and thus \( \sqrt{\Delta} \) is an even number if \( \Delta \) is a perfect square.
    • The numerator in the quadratic formula becomes \( -2k \pm \sqrt{\Delta} \). Since \( \sqrt{\Delta} \) is even, the entire numerator is even, ensuring that division by 2 results in an integer.

  2. When \( b \) is an Odd Integer:

    • Here, \( b \) can be expressed as \( b = 2k + 1 \).
    • The discriminant \( \Delta \) is \( \Delta = (2k + 1)^2 - 4c \).
    • Observe that \( (2k + 1)^2 \) is an odd number (as the square of an odd number is odd) and \( -4c \) is even.
    • Therefore, \( \Delta \) is odd (since the difference of an odd number and an even number is odd).
    • If \( \Delta \) is a perfect square and odd, then \( \sqrt{\Delta} \) is also an odd number.
    • The numerator in the quadratic formula becomes \( -(2k + 1) \pm \sqrt{\Delta} \). The addition or subtraction of an odd number from another odd number results in an even number.
    • Hence, the numerator is even, and division by 2 results in an integer.

In both scenarios—whether \( b \) is even or odd—if the discriminant \( \Delta \) is a perfect square, the numerator of the quadratic formula is an even number. This even numerator ensures that the division by 2 yields integer roots. Therefore, the roots of the monic quadratic expression are integers if and only if \( \Delta \) is a perfect square.

The integer nature of the roots in a monic quadratic expression with integer coefficients is directly tied to the discriminant being a perfect square. This theorem illustrates that regardless of whether \( b \) is even or odd, if \( \Delta \) is a perfect square, the structure of the quadratic formula guarantees that the roots are integers.

Analyzing a pair of quadratic equations

When examining two quadratic equations simultaneously, the sum of their discriminants or sometimes their product can provide insights into the collective nature of their roots. Let's analyze this relationship.

First read this example to clearly understand the theory of this section.

Example

Considering two quadratic equations:

  1. \( ax^2 + bx + c = 0 \) with discriminant \( \Delta_1 = b^2 - 4ac \).
  2. \( ax^2 + bx - c = 0 \) with discriminant \( \Delta_2 = b^2 + 4ac \).

Assuming that \( b \neq 0 \), we analyze the sum of their discriminants:

\( \Delta_1 + \Delta_2 = (b^2 - 4ac) + (b^2 + 4ac) \)
\(\implies\)\( \Delta_1 + \Delta_2 = 2b^2 \)

Since \( b^2 \) is positive for any non-zero \( b \), \( \Delta_1 + \Delta_2 > 0 \). This ensures that the sum of the discriminants is always positive.

The fact that \( \Delta_1 + \Delta_2 > 0 \) indicates that at least one of \( \Delta_1 \) or \( \Delta_2 \) is positive. Consequently, at least one of the quadratic equations has real and distinct roots. Therefore, collectively, there are at least two real and distinct roots between the two equations.

I. Sum of discriminants of two quadratic equations is known

  1. Sum of discriminants is non-negative

    If \( D_1 \) and \( D_2 \) are the discriminants of two quadratic equations and \( D_1 + D_2 \geq 0 \) then at least one of \( D_1 \) and \( D_2 \) is non-negative. There are two subcases:

    • Case1: \(D_1+D_2>0\): It can be concluded that at least one of \(D_1\) or \(D_2\) is positive, implying that collectively they have at least 2 distinct real roots.
    • Case 2: \(D_1+D_2=0\): It can be concluded that either exactly one of \(D_1\) or \(D_2\) is positive or both are zero. Whatever be the case this again ensures that there are atleast two real and distinct roots.

    It can be finally concluded that when \( D_1 + D_2 \geq 0 \), then there are collectively at least 2 real and distinct roots.

  2. Sum of Discriminants is Negative

    If the sum of the discriminants \( D_1 + D_2 < 0 \), it means that the negative discriminant has a larger magnitude than the positive one, or both are negative. This indicates that at least one of the quadratic equations has complex (non-real) roots because for real roots to occur, the discriminant must be non-negative.

II. Product of discriminants of two quadratic equations is known

If \( D_1 \) is the discriminant of \( ax^2 + bx + c = 0 \) and \( D_2 \) of \( a_2x^2 + b_2x + c_2 = 0 \), then:

(i) If \( D_1, D_2 < 0 \), then at least one of \( D_1 \) or \( D_2 \) is negative, indicating at least one equation has complex roots.
(ii) If \( D_1, D_2 \geq 0 \), there are three subcases:

  • Case I: If both \( D_1 \) and \( D_2 \) are positive, each equation has two distinct real roots, totaling four real roots combined.
  • Case II: If both \( D_1 \) and \( D_2 \) are zero, each equation has one real repeated root, giving two real roots that are the same for each equation.
  • Case III: If one \( D \) is positive and the other is zero, then one equation has two distinct real roots, and the other has a repeated real root, leading to three distinct real roots in total.

In summary, the discriminants of quadratic equations determine the nature of their roots. A non-negative discriminant indicates real roots, while a negative one implies complex roots. By analyzing the sum or product of discriminants, we can infer the possible combinations of roots for a pair of quadratic equations.