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Chapter 1: Problem 14

Let \(p\) and \(q\) be roots of the equation \(x^{2}-2 x+A=0\) and let \(r\) and \(s\) be the roots of the equation \(x^{2}-18 x+B=0\). If \(p

Short Answer

Expert verified
The values of \(A\) and \(B\) are \(-3\) and \(77\), corresponding to option d.

Step by step solution

01

Understanding the Problem

We are given two quadratic equations: \(x^2 - 2x + A = 0\) and \(x^2 - 18x + B = 0\) with roots \(p, q\) and \(r, s\) respectively. We know these roots form an arithmetic progression with \(p < q < r < s\). We need to find the values of \(A\) and \(B\).
02

Use Vieta's Formulas

By Vieta's formulas for \(x^2 - 2x + A = 0\), we have \(p + q = 2\) and \(pq = A\). For \(x^2 - 18x + B = 0\), \(r + s = 18\) and \(rs = B\).
03

Set the Arithmetic Progression Equations

Since \(p, q, r, s\) are in arithmetic progression, we have the relationships: \(q = p + d\), \(r = p + 2d\), and \(s = p + 3d\). By substituting these into Vieta's sums, we can express \(p, q, r, s\) in terms of \(d\) and use the conditions: \(p + q = 2\) and \(r + s = 18\).
04

Solve for Common Difference \(d\)

From the equation \(p + q = 2\), replacing \(q\) by \(p + d\), we get \(p + (p + d) = 2\). Therefore, \(2p + d = 2\). Similarly, from \(r + s = 18\), substituting \(r\) and \(s\) gives us \((p + 2d) + (p + 3d) = 18\), which simplifies to \(2p + 5d = 18\).
05

Solve the System of Equations

We now solve the two equations: \(2p + d = 2\) and \(2p + 5d = 18\). Subtract the first equation from the second to eliminate \(2p\): \((2p + 5d) - (2p + d) = 18 - 2\), which simplifies to \(4d = 16\) leading to \(d = 4\). With \(d = 4\), substitute back to find \(p\): \(2p + 4 = 2\), thus \(2p = -2\) and \(p = -1\).
06

Calculate the Remaining Roots

With \(p = -1\) and \(d = 4\), we find \(q = p + d = 3\), \(r = p + 2d = 7\), and \(s = p + 3d = 11\). Check these satisfy the conditions \(p < q < r < s\).
07

Find Values of \(A\) and \(B\)

Use the products from Vieta's formulas: \(pq = A = (-1)(3) = -3\) and \(rs = B = (7)(11) = 77\). Therefore, \(A = -3\) and \(B = 77\).
08

Verify and Select the Correct Option

Cross-reference the found values \((A = -3, B = 77)\) with the given options. The correct choice is d. \(-3, 77\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vieta's formulas
Vieta's formulas are a powerful tool used mainly to relate the coefficients of a polynomial to sums and products of its roots. When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), Vieta's formulas state that:
  • The sum of the roots \((\alpha + \beta) = -\frac{b}{a}\)
  • The product of the roots \((\alpha \beta) = \frac{c}{a}\)
For the equation \(x^2 - 2x + A = 0\), applying Vieta's formulas gives us:
  • Sum of roots, \(p + q = 2\)
  • Product of roots, \(pq = A\)
Similarly, for the equation \(x^2 - 18x + B = 0\), Vieta's provides:
  • Sum of roots, \(r + s = 18\)
  • Product of roots, \(rs = B\)
These relationships help us understand how the roots are connected to the coefficients. Using Vieta's formulas is essential when considering the structure and behavior of polynomial roots.
solving quadratic equations
Solving quadratic equations is a fundamental aspect of algebra. A quadratic equation is in the form \(ax^2 + bx + c = 0\). The solutions, or "roots," to this equation, can be found through several methods including:
  • Factoring: If the quadratic can be expressed as a product of binomials.
  • Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
  • Completing the Square: Rewriting the equation to make the left side a perfect square trinomial.
  • Graphical Method: Finding the x-intercepts of the parabola formed by the equation.
In our case, we specifically looked into the equations \(x^2 - 2x + A = 0\) and \(x^2 - 18x + B = 0\). Typically, we apply the quadratic formula or factoring for an immediate solution, but here we connect back to the condition of roots being in an arithmetic progression with Vieta鈥檚 formulas as our guide.
system of equations
A system of equations involves finding the values of variables that satisfy multiple equations simultaneously. In this exercise, we derive a system of two equations based on the conditions provided:
  • \(2p + d = 2\) (from the condition that \(p\) and \(q = p + d\) sum to 2)
  • \(2p + 5d = 18\) (from the condition that \(r = p + 2d\) and \(s = p + 3d\) sum to 18)
We solve this system to find the common difference \(d\) and the root \(p\). By manipulating these equations, we determine:
  • After subtraction and simplification, \(4d = 16\) or \(d = 4\)
  • Substituting \(d = 4\) back, we find \(p = -1\)
This systematic approach allows us to understand how the roots progress and aids in determining the actual values of \(A\) and \(B\) using further calculations.
quadratic roots in arithmetic progression
When quadratic roots are said to be in arithmetic progression, they follow the sequence pattern: each term is equal to the previous term plus a constant 'difference'. For sets \(p < q < r < s\) in arithmetic progression:
  • \(q = p + d\)
  • \(r = p + 2d\)
  • \(s = p + 3d\)
With given sequences, each series term increases by the same amount. To align with Vieta鈥檚 sums:
  • \(p + q = 2 \Rightarrow p + (p + d) = 2\)
  • \(r + s = 18 \Rightarrow (p + 2d) + (p + 3d) = 18\)
These constraints help us define linkages between roots' values and the quadratic coefficients through both direct sums and differences. It results in a precise identification of roots for deeper algebraic exploration and practical solution approaches. This structured sequence helps to solve problems in mathematical settings efficiently.

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Most popular questions from this chapter

If \(a, b\) and \(c\) are real numbers such that \(a^{2}+b^{2}+c^{2}=1\), then \(a b\) \(+b c+c a\) lies in the interval a. \([1 / 2,2]\) b. \([-1,2]\) c. \([-1 / 2,1]\) d. \([-1,1 / 2]\)

If \(a, b, c, d \in R\), then the cquation \(\left(x^{2}+a x-3 b\right)\left(x^{2}-c x+b\right)\) \(\left(x^{2}-d x+2 b\right)=0\) has a. 6 real roots b. at least 2 real roots c. 4 real roots d. 3 real roots

The number of roots of the equation \(\sqrt{x}-2\left(x^{2}-4 x+3\right)=0\) is a. three b. four c. one d. two

If roots of an equation \(x^{x}-1=0\) are \(1, a_{1}, a_{2}, \ldots, a_{n-1}\), then the value of \(\left(1-a_{1}\right)\left(1-a_{2}\right)\left(1-a_{3}\right) \cdots\left(1-a_{n-1}\right)\) will be a. \(n\) b. \(n^{2}\) c. \(n^{\prime \prime}\) d. 0

If both roots of the equation \(a x^{2}+x+c-a=0\) are imaginary and \(c>-1\), then a. \(3 a>2+4 c\) b. \(3 a<2+4 c\) c. \(c
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