Question

The roots of the equation \(x^{2}-p x+q=0\) are consecutive integers. Find the discriminant of the equation. (1) 1 (2) 2 (3) 3 (4) 4

Short answer

Answer: The discriminant of the given equation is 1.

Step-by-step solution

Let the consecutive integers be \(n\) and \(n+1\). Then, we know that the sum of roots is \(n + (n+1) = 2n+1\) and the product of roots is \(n(n+1)\). #Step 2: Equating Sum and Product of Roots#

We know that for our given equation \(x^2 - px + q = 0\), the sum of roots is \(p\) and the product of roots is \(q\). Thus, we can equate these expressions for the consecutive integers to find the values of \(p\) and \(q\). Sum of roots: \(p = 2n + 1\) Product of roots: \(q = n(n + 1)\) #Step 3: Finding the Discriminant#

Now we have to find the discriminant of the equation \(x^2 - px + q = 0\). The discriminant is given by the formula \(D = b^2 - 4ac\), where \(a = 1\), \(b = -p\), and \(c = q\). For our given equation, \(D = p^2 - 4q\). #Step 4: Substituting the Values of p and q#

We substitute the expressions of \(p\) and \(q\) found in Step 2 into the discriminant formula obtained in Step 3. \(D = (2n + 1)^2 - 4(n(n+1))\) #Step 5: Simplifying the Expression#

Now, let's simplify the expression for the discriminant: \(D = (2n^2 + 4n + 1) - 4(n^2 + n)\) \(D = 2n^2 + 4n + 1 - 4n^2 - 4n\) \(D = -2n^2 + 1\) #Step 6: Consecutive Integer Property#

Since the roots are consecutive integers, the discriminant must be a perfect square. It means that the expression \(-2n^2 + 1\) must be equal to a perfect square. When \(n = 0\), \(D = 1\). And when \(n=1\), \(D= -1\). Since the values of \(n\) are integers, the smallest value for D possible is 1. Thus, the discriminant of the equation is 1 (Option 1).

Key concepts

Roots of Quadratic Equation

Understanding the roots of a quadratic equation can be crucial in solving various mathematical problems. A quadratic equation typically takes the form of
\( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \eq 0 \). The solutions to this equation, known as the roots, can be real or complex numbers. These roots are the values of \( x \) that make the equation true.
When the roots are consecutive integers, it implies a specific relationship between them. For instance, if one root is \( n \), the other is \( n+1 \). This property simplifies the process of finding the discriminant, which is the part of the quadratic formula \( \frac{-b\pm\sqrt{b^2-4ac}}{2a} \) that is under the square root sign, \( \sqrt{b^2-4ac} \)—denoted as \( D \). The discriminant gives us valuable information about the nature of the roots, indicating whether they are real and distinct, real and same, or complex. In this specific exercise, we are guided to find the discriminant based on the fact that the roots are consecutive integers.

Consecutive Integers

The concept of consecutive integers is straightforward: they are integers that follow one another, without any gaps in between. For example, 1, 2, 3 are consecutive integers. When dealing with quadratic equations, if you know the roots are consecutive, you’re dealing with a sequence of numbers like \( n \) and \( n+1 \).
This knowledge allows us to derive the sum and product of roots easily. Since \( p \) represents the sum of the roots in the quadratic equation \( x^2 - px + q = 0 \), for consecutive integers, this sum is always an odd number, \( 2n+1 \). Knowing that the roots are consecutive offers a strategy for quickly determining the structure of the solution and provides an interesting property that can be used to solve for the discriminant—as we’ve seen in the exercise.
Remember that for any quadratic equation, if the roots are integers, the discriminant will always be a perfect square.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation hold significant importance in simplifying the process of solving the equation. These can be found using the coefficients of the equation \( ax^2 + bx + c = 0 \). For the sum, \( -\frac{b}{a} \) gives the sum of the roots, while for the product, \( \frac{c}{a} \) provides the answer.
In our exercise, \( p \) and \( q \) are derived from setting the general forms of sum and product equal to those established by the consecutive integers. It is vital to comprehend that this relationship between the sum and product of roots, and the equation's coefficients, is always true, irrespective of whether the roots are integers, real, or complex numbers.
Understanding the application of these relationships allows us to find the discriminant effectively without necessarily finding the exact roots of the equation. This illustration serves as a lay foundation for dealing with more complex problems involving quadratic equations and their roots.