Location of Roots
Suppose we want to determine the location of the number 3 with respect to the roots of the quadratic expression \(3x^2 + 4x  1\), tthat is, we want to determine whether 3 lies to the left of roots(less than both roots), between both roots, or right of roots(greater than both roots). An obvious way would be the following:

Calculate the Roots: Find the roots of \(3x^2 + 4x  1\). The roots can be found using the quadratic formula:
\[ x = \frac{b \pm \sqrt{b^2  4ac}}{2a} \]where \(a = 3\), \(b = 4\), and \(c = 1\).
Let's calculate the roots and then determine the position of the number 3 relative to these roots.
The roots of the quadratic equation \(3x^2 + 4x  1 = 0\) are approximately \(2/3 + \sqrt{7}/3\) and \(\sqrt{7}/3  2/3\).

Compare with 3: These values are both less than 3. Since both roots are less than 3, the number 3 lies to the right of both roots. This means, in the context of the quadratic expression \(3x^2 + 4x  1\), the value 3 does not fall between the roots, but rather is located to the right of them on the number line.
Such direct calaculation of roots and then making a comparision can be cumbersome and for certain form of coefficients it can also be cumbersome. So we need an indirect way. As it turns out it is quite intuitive and simple.
k lies to the LEFT of both roots
Given a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients \( a \), \( b \), and \( c \), and a real number \( k \), the conditions under which \( k \) lies to the left of both roots of the quadratic equation are:
 The discriminant \(\Delta \geq 0\).
 \(a\) and the value of the quadratic expression at \( k \) have the same sign, i.e., \( af(k) > 0 \).
 \( k \) is less than the xcoordinate of the vertex of the parabola, i.e., \( k < \frac{b}{2a} \).
Proof
To prove this theorem, we consider the nature of a quadratic equation and its graph, which is a parabola.

Discriminant Condition (\(\Delta \geq 0\)):
 A nonnegative discriminant (\(\Delta \geq 0\)) ensures that the quadratic equation has real roots.
 If \(\Delta < 0\), the quadratic equation has no real roots, and the condition about \( k \) being to the left of both roots becomes meaningless.

Value of \( f(k) \) ( \( af(k) > 0 \) ):
 The sign of \( f(k) \) indicates on which side of the xaxis the graph of the quadratic expression lies at \( k \).
 For \( k \) to be to the left of both roots, \( f(k) \) must be above the xaxis if \( a > 0 \) (parabola opening upwards) or below the xaxis if \( a < 0 \) (parabola opening downwards). In short we can say that \(a\) and \(f(k)\) have the same sign. Mathmatically this can be written as \(af(k)>0\).

Position Relative to Vertex ( \( k < \frac{b}{2a} \) ):
 The xcoordinate of the vertex of the parabola is given by \( \frac{b}{2a} \), whaich is also the midpoint of both roots.
 For \( k \) to be to the left of both roots, it must be less than the xcoordinate of the vertex.
 This condition ensures that \( k \) is positioned to the left of the axis of symmetry of the parabola, which is essential for \( k \) to be to the left of both roots when they are real and distinct.
In conclusion, when a quadratic expression \( f(x) = ax^2 + bx + c \) has real coefficients and a real number \( k \) satisfies the conditions \(\Delta \geq 0\), \( af(k) > 0 \), and \( k < \frac{b}{2a} \), then \( k \) lies to the left of both roots of the quadratic equation.
k lies to the RIGHT of both roots
Given a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients \( a \), \( b \), and \( c \), and a real number \( k \), the conditions under which \( k \) lies to the right of both roots of the quadratic equation are:
 The discriminant \(\Delta \geq 0\).
 \(a\) and the value of the quadratic expression at \( k \) have the same sign, i.e., \( af(k) > 0 \).
 \( k \) is greater than the xcoordinate of the vertex of the parabola, i.e., \( k > \frac{b}{2a} \).
Proof
To prove this theorem, we again consider the nature of a quadratic equation and its graph, which is a parabola.
 Discriminant Condition (\(\Delta \geq 0\)):
 A nonnegative discriminant (\(\Delta \geq 0\)) ensures that the quadratic equation has real roots.

A negative discriminant (\(\Delta < 0\)) implies no real roots, making the condition about \( k \) being to the right of both roots irrelevant.

Value of \( f(k) \) ( \( af(k) > 0 \) ):
 The sign of \( f(k) \) at the point \( k \) determines on which side of the xaxis the graph lies at \( k \).
 If \( a > 0 \) (upward opening parabola), \( f(k) > 0 \) suggests \( k \) is outside the interval between the roots. If \( a < 0 \) (downward opening parabola), \( f(k) < 0 \) suggests the same.

Thus, \( af(k) > 0 \) ensures \( k \) lies outside the interval of the roots on the graph corresponding to the orientation of the parabola.

Position Relative to Vertex ( \( k > \frac{b}{2a} \) ):
 The vertex's xcoordinate is \( \frac{b}{2a} \), which is the midpoint between the roots for a quadratic equation with real and distinct roots.
 For \( k \) to be to the right of both roots, it must be greater than the vertex's xcoordinate.
 This condition ensures \( k \) is positioned to the right of the parabola's axis of symmetry, crucial for \( k \) to be to the right of both roots when they are real and distinct.
In conclusion, when a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients has a real number \( k \) that satisfies the conditions \(\Delta \geq 0\), \( af(k) > 0 \), and \( k > \frac{b}{2a} \), then \( k \) lies to the right of both roots of the quadratic equation.
k lies BETWEEN the roots.
Given a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients \( a \), \( b \), and \( c \), and a real number \( k \), the condition under which \( k \) lies between the roots of the quadratic equation is \( af(k) < 0 \).
Proof
To prove this theorem, we consider the nature of a quadratic equation and its graph, which is a parabola.
Value of \( f(k) \) ( \( af(k) < 0 \) ):
 The sign of \( f(k) \) at the point \( k \) determines on which side of the xaxis the graph of the quadratic expression lies at \( k \).
 If \( a > 0 \) (upward opening parabola), \( f(k) < 0 \) suggests \( k \) is within the interval between the roots. If \( a < 0 \) (downward opening parabola), \( f(k) > 0 \) suggests the same.
 Therefore, \( af(k) < 0 \) ensures that \( k \) lies between the roots on the graph of the parabola.
In conclusion, when a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients has a real number \( k \) that satisfies the condition \( af(k) < 0 \), then \( k \) lies between the roots of the quadratic equation.
Note
The discriminant condition (\(\Delta \geq 0\)) is not necessary here because the situation where the discriminant is negative (no real roots) or zero (roots coincide) inherently does not allow for a real number \( k \) to lie between the roots. The sole condition \( af(k) < 0 \) suffices to determine that \( k \) lies between the roots when the roots are real and distinct.
You are correct. The condition for exactly one root of the quadratic equation \( f(x) = ax^2 + bx + c \) to lie in the interval \([k_1, k_2]\) can indeed be simplified to:
Exactly one root lies BETWEEN two real numbers
Given a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients \( a \), \( b \), and \( c \), and two real numbers \( k_1 \) and \( k_2 \) (where \( k_1 \leq k_2 \)), the condition under which exactly one root of the quadratic equation lies in the interval \([k_1, k_2]\) is: \( f(k_1) \cdot f(k_2) < 0 \).
Proof
Sign of \( f(k_1) \) and \( f(k_2) \):
 The sign of \( f(k) \) at the points \( k_1 \) and \( k_2 \) indicates on which side of the xaxis the graph of the quadratic expression lies at these points.
 If \( f(k_1) \) and \( f(k_2) \) have opposite signs, it means the graph of the parabola crosses the xaxis within the interval \([k_1, k_2]\), indicating exactly one root lies in this interval.
 This condition stems from the Intermediate Value Theorem, which states that if a continuous function (such as a quadratic function) has values of opposite signs at two points, then it must cross the xaxis at least once between those points.
In conclusion, the condition \( f(k_1) \cdot f(k_2) < 0 \) is sufficient to determine that exactly one root of the quadratic equation \( f(x) = ax^2 + bx + c \) lies in the interval \([k_1, k_2]\).
Both roots lie between two real numbers.
Given a quadratic expression \( f(x) = ax^2 + bx + c \) with real coefficients \( a \), \( b \), and \( c \), and two real numbers \( k_1 \) and \( k_2 \) (where \( k_1 \leq k_2 \)), the conditions under which both roots of the quadratic equation lie in the interval \([k_1, k_2]\) are:
 The discriminant \(\Delta \geq 0\).
 \(a\) and the value of the quadratic expression at \( k_1 \) have the same or zero sign, i.e., \( af(k_1) \geq 0 \).
 \(a\) and the value of the quadratic expression at \( k_2 \) have the same or zero sign, i.e., \( af(k_2) \geq 0 \).
 \( k_1 \) is less than or equal to, and \( k_2 \) is greater than or equal to the xcoordinate of the vertex of the parabola, i.e., \( k_1 \leq \frac{b}{2a} \leq k_2 \).
Proof

Discriminant Condition (\(\Delta \geq 0\)):
 The discriminant \(\Delta = b^2  4ac\) determines the nature of the roots. A nonnegative discriminant ensures that the quadratic equation has real roots.

Value of \( f(k_1) \) and \( f(k_2) \) ( \( af(k_1) \geq 0 \) and \( af(k_2) \geq 0 \) ):
 The sign of \( f(k) \) at \( k_1 \) and \( k_2 \) indicates on which side of the xaxis the graph of the quadratic expression lies at these points.
 For both roots to be within or on the boundary of the interval \([k_1, k_2]\), the function value at these points must be nonnegative when multiplied by \( a \), ensuring the graph is not below the xaxis at these points for an upwardopening parabola, or not above the xaxis for a downwardopening parabola.

Position Relative to Vertex ( \( k_1 \leq \frac{b}{2a} \leq k_2 \) ):
 The vertex's xcoordinate, \( \frac{b}{2a} \), is the midpoint between the roots for a quadratic equation with real and distinct roots.
 For both roots to be within the interval \([k_1, k_2]\), the vertex must lie between or at \( k_1 \) and \( k_2 \).
 This ensures that the interval \([k_1, k_2]\) encompasses the entire spread of the roots.
In conclusion, the satisfaction of these four conditions ensures that both roots of the quadratic equation \( f(x) = ax^2 + bx + c \) lie within the interval \([k_1, k_2]\).