Problem 23
Question
If the roots of \(p x^{2}+q x+2=0\) are reciprocal of each other, then find \(p\).
Step-by-Step Solution
Verified Answer
The value of \(p\) is 2.
1Step 1: Formulation of the condition
Given that the roots of the equation are reciprocals, if one root is \(r\), then the other is \(1/r\). Hence the product of the roots \(r \cdot (1/r) = 1\).
2Step 2: Usage of Vieta's formula
According to Vieta's formula, the product of the roots for the equation \(p x^{2}+q x+2=0\) can be represented as \(c/a\), that is, \(2/p\).
3Step 3: Equate and solve for p
Equating our known factors, we get \(1 = 2/p\), thus, by solving for \(p\), we find \(p = 2\).
Key Concepts
Vieta's FormulasProduct of RootsQuadratic Equations
Vieta's Formulas
Vieta's formulas provide a powerful connection between the coefficients of a polynomial and its roots. In the realm of quadratic equations, these formulas particularly simplify the process of understanding the relationship between the roots and coefficients.
For a quadratic equation of the form \(ax^{2} + bx + c = 0\), Vieta's formulas state that the sum of the roots (\(r_1 + r_2\)) is equal to \( -\frac{b}{a}\) and the product of the roots (\(r_1 \cdot r_2\)) is \( \frac{c}{a}\). This implies that without explicitly solving for the roots, one can deduce vital information regarding their properties.
Applying these formulas can instantly reveal if the roots have certain characteristics, such as being reciprocal pairs—as is the case in the exercise presented. When the roots are reciprocal, meaning one root is \( \frac{1}{r}\) and the other one is \(r\), the product will invariably be 1. This illustrates how Vieta's can affirm a root relationship in a straightforward manner.
For a quadratic equation of the form \(ax^{2} + bx + c = 0\), Vieta's formulas state that the sum of the roots (\(r_1 + r_2\)) is equal to \( -\frac{b}{a}\) and the product of the roots (\(r_1 \cdot r_2\)) is \( \frac{c}{a}\). This implies that without explicitly solving for the roots, one can deduce vital information regarding their properties.
Applying these formulas can instantly reveal if the roots have certain characteristics, such as being reciprocal pairs—as is the case in the exercise presented. When the roots are reciprocal, meaning one root is \( \frac{1}{r}\) and the other one is \(r\), the product will invariably be 1. This illustrates how Vieta's can affirm a root relationship in a straightforward manner.
Product of Roots
Understanding the product of roots in a quadratic equation is essential for uncovering the relationship between the quadratic coefficients and its roots. Following Vieta's formulas, for any quadratic equation \(ax^{2} + bx + c = 0\), if we denote the roots as \(r_1\) and \(r_2\), the product of the roots will always equal \(\frac{c}{a}\).
This means that if you're given that the product of the roots is a certain value—like 1 in the case of reciprocal roots—you can instantly infer a direct proportionality between the constant term \(c\) and the leading coefficient \(a\).
So, given the quadratic equation \(px^{2} + qx + 2 = 0\) with reciprocal roots, the product \(r_1 \cdot r_2\) equates to \(\frac{2}{p}\). If the product of reciprocal roots is known to be 1, then \(\frac{2}{p} = 1\), which leads to a straightforward calculation to find the value of \(p\). This aspect of quadratic relationships can be incredibly helpful in solving equations without needing to find the exact roots.
This means that if you're given that the product of the roots is a certain value—like 1 in the case of reciprocal roots—you can instantly infer a direct proportionality between the constant term \(c\) and the leading coefficient \(a\).
So, given the quadratic equation \(px^{2} + qx + 2 = 0\) with reciprocal roots, the product \(r_1 \cdot r_2\) equates to \(\frac{2}{p}\). If the product of reciprocal roots is known to be 1, then \(\frac{2}{p} = 1\), which leads to a straightforward calculation to find the value of \(p\). This aspect of quadratic relationships can be incredibly helpful in solving equations without needing to find the exact roots.
Quadratic Equations
Quadratic equations are mathematical expressions of the second degree that take the general form \(ax^{2} + bx + c = 0\). They are fundamental elements within algebra and are characterized by their 'U'-shaped graph known as a parabola.
When it comes to solving quadratic equations, there are multiple techniques, such as factoring, using the quadratic formula, completing the square, or graphing. However, one must not overlook the importance of properties such as the product and sum of the roots, provided by Vieta's formulas, which offer an insightful way to analyze the equation without explicitly finding its solutions.
A fascinating property of quadratic equations is their symmetry, which is apparent in scenarios where the roots are reciprocals of each other. The symmetry conveys that one can deduce certain coefficients of the equation by knowing this particular root behavior, highlighting the beautifully interconnected nature of the equation's parts.
When it comes to solving quadratic equations, there are multiple techniques, such as factoring, using the quadratic formula, completing the square, or graphing. However, one must not overlook the importance of properties such as the product and sum of the roots, provided by Vieta's formulas, which offer an insightful way to analyze the equation without explicitly finding its solutions.
A fascinating property of quadratic equations is their symmetry, which is apparent in scenarios where the roots are reciprocals of each other. The symmetry conveys that one can deduce certain coefficients of the equation by knowing this particular root behavior, highlighting the beautifully interconnected nature of the equation's parts.
Other exercises in this chapter
Problem 21
If \(\sin \alpha\) and \(\cos \alpha\) are roots of the equation \(p x^{2}+q x+r=0\), then show that \(p^{2}-q^{2}+2 p r=0\).
View solution Problem 22
If one root of the equation \(5 x^{2}+13 x+k=0\) is reciprocal of other, then find the value of \(k\).
View solution Problem 24
Find the condition that the roots of the equation \(a x^{2}+b x+c=0\) be such that i. One root is \(n\) times the other. ii. One root is three times the other.
View solution Problem 25
If the roots of the equation \(a x^{2}+b x+c=0\) are of the form \(\frac{k+1}{k}\) and \(\frac{k+2}{k+1}\), prove that \((a+b+c)^{2}=b^{2}-4 a c\)
View solution