Interval in which roots lie

When studying quadratic equations, it is often useful to determine the interval within which the roots of the equation lie. This can provide insights into the behavior of the function represented by the equation without necessarily finding the exact roots.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form:

$$ ax^2 + bx + c = 0 $$

where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ).

The roots of a quadratic equation are the values of ( x ) that satisfy the equation. These roots can be real or complex, and there can be two distinct roots, one repeated root, or no real roots at all.

Determining the Interval for Real Roots

To find the interval in which the real roots of a quadratic equation lie, we can use the following properties:

Sign of the Leading Coefficient: The sign of ( a ) determines the direction of the parabola. If ( a > 0 ), the parabola opens upwards, and if ( a < 0 ), it opens downwards.
Discriminant: The discriminant of a quadratic equation, ( \Delta = b^2 - 4ac ), determines the nature of the roots. If ( \Delta > 0 ), there are two distinct real roots; if ( \Delta = 0 ), there is one repeated real root; and if ( \Delta < 0 ), there are no real roots.
Vertex: The vertex of the parabola is given by ( x = -\frac{b}{2a} ). The ( y )-coordinate of the vertex is the minimum or maximum value of the quadratic function, depending on the sign of ( a ).
Axis of Symmetry: The line ( x = -\frac{b}{2a} ) is the axis of symmetry of the parabola. The roots are equidistant from this line if they are real.

Table of Properties

Property	Description	Impact on Roots Interval
Sign of ( a )	Determines the direction of the parabola	Defines if the interval is above or below the vertex
Discriminant ( \Delta )	Determines the nature of the roots	Indicates if real roots exist and if they are distinct
Vertex	Gives the minimum or maximum value of the function	Serves as a boundary for the interval of roots
Axis of Symmetry	Line equidistant from the roots if they are real	Helps locate the roots on the ( x )-axis

Formulas

The roots of a quadratic equation, when they are real, can be found using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{\Delta}}{2a} $$

This formula gives the exact values of the roots when they are real. However, to find the interval in which the roots lie, we can use the properties mentioned above without necessarily computing the roots explicitly.

Examples

Let's consider some examples to illustrate how to determine the interval in which the roots of a quadratic equation lie.

Example 1: Two Distinct Real Roots

Consider the quadratic equation ( x^2 - 5x + 6 = 0 ).

The leading coefficient ( a = 1 ) is positive, so the parabola opens upwards.
The discriminant ( \Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0 ), so there are two distinct real roots.
The vertex is at ( x = -\frac{-5}{2(1)} = 2.5 ), and since ( a > 0 ), this is the minimum point.
The axis of symmetry is ( x = 2.5 ).

Since the parabola opens upwards and the vertex is the minimum point, we know that the roots lie to the left and right of ( x = 2.5 ). The exact roots can be found using the quadratic formula, but for the interval, we can say that one root lies in ( (-\infty, 2.5) ) and the other in ( (2.5, +\infty) ).

Example 2: One Repeated Real Root

Consider the quadratic equation ( x^2 - 4x + 4 = 0 ).

The leading coefficient ( a = 1 ) is positive, so the parabola opens upwards.
The discriminant ( \Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0 ), so there is one repeated real root.
The vertex and the repeated root are at ( x = -\frac{-4}{2(1)} = 2 ).

Since there is one repeated root, the interval in which this root lies is simply the point ( x = 2 ).

Example 3: No Real Roots

Consider the quadratic equation ( x^2 + x + 1 = 0 ).

The leading coefficient ( a = 1 ) is positive, so the parabola opens upwards.
The discriminant ( \Delta = (1)^2 - 4(1)(1) = 1 - 4 = -3 < 0 ), so there are no real roots.

In this case, there is no interval for real roots since they do not exist. The roots are complex.

By understanding these properties and applying them to a given quadratic equation, one can determine the interval in which the roots lie without necessarily computing the exact roots. This knowledge can be particularly useful in graphing quadratic functions and solving inequalities.