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Chapter 4: Problem 9

Let \(a, b, c\) be positive real numbers, such that \(b x^{2}+\) \(\left(\sqrt{(a+c)^{2}+4 b^{2}}\right) x+(a+c) \geq 0, \forall x \in R\), then \(a, b, c\) are in: (A) G.P. (B) A.P. (C) H.P. (D) None of these

Short Answer

Expert verified
Option B: A.P. (Arithmetic Progression)

Step by step solution

01

Identify the Expression

The given inequality is \(b x^2 + \sqrt{(a+c)^2 + 4b^2} x + (a+c) \geq 0\) for all \(x \in \mathbb{R}\). We need to consider this expression as a quadratic in \(x\): \(A = b\), \(B = \sqrt{(a+c)^2 + 4b^2}\), and \(C = (a+c)\).
02

Condition for Non-Negativity

For the quadratic expression to be non-negative for all real \(x\), its discriminant must be less than or equal to zero. The discriminant \(\Delta\) of the quadratic \(Ax^2 + Bx + C\) is calculated as \(\Delta = B^2 - 4AC\). Here, \(A = b, B = \sqrt{(a+c)^2 + 4b^2}, C = a+c\).
03

Calculate the Discriminant

Substitute the values of \(A, B, C\) into the discriminant formula: \[\Delta = \left(\sqrt{(a+c)^2 + 4b^2}\right)^2 - 4b(a+c)\]. Simplify it to \[\Delta = (a+c)^2 + 4b^2 - 4b(a+c)\].
04

Simplify the Discriminant

Rewrite the expression: \((a+c)^2 - 4b(a+c) + 4b^2 = (a+c-2b)^2\). Thus, \(\Delta = (a+c-2b)^2\).
05

Analyze the Discriminant for Non-Positivity

For the quadratic to be always non-negative, \((a+c-2b)^2 \leq 0\). Since a square of any real number is non-negative, \((a+c-2b)^2 = 0\), hence \(a+c = 2b\).
06

Conclusion

From \(a+c = 2b\), it follows that \(a, b, c\) must be in Arithmetic Progression (A.P.) where \((a, b, c) = \left(a, \frac{a+c}{2}, c\right)\).

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Key Concepts

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

Arithmetic Progression (A.P.)
An arithmetic progression, often abbreviated as A.P., is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2 because each term increases by 2.
  • To denote A.P., we generally use the form: a, a+d, a+2d,..., where \( d \) is the common difference.
  • A property of numbers in an A.P. is that the middle term is the average of the terms on either side.
In the problem statement, the condition given was \( a+c = 2b \). This implies \( b \) is exactly in the middle of \( a \) and \( c \), verifying the nature of an arithmetic progression: \( a, b, c \) are in A.P. because the difference between \( b \) and \( a \) is equal to the difference between \( c \) and \( b \).
Discriminant analysis
The discriminant of a quadratic expression helps determine the nature of the roots of the quadratic equation. Given a quadratic equation \( Ax^2 + Bx + C = 0 \), its discriminant \( \Delta \) is calculated as \( \Delta = B^2 - 4AC \). Here is how the discriminant guides us:
  • If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
  • If \( \Delta = 0 \), it has exactly one real root or a repeated root.
  • If \( \Delta < 0 \), the roots are complex and do not exist on the real number line.
In our problem, for the quadratic to be non-negative for all real numbers, its discriminant must satisfy \( \Delta \leq 0 \). The solution set manipulated the discriminant \((a+c-2b)^2\) to zero, aligning perfectly with \( \Delta = 0\). This condition ensures no real roots cut the x-axis, backing up our inequality.
Quadratic Expressions
Quadratic expressions are central in algebra and take the form \( Ax^2 + Bx + C \). These expressions graph as parabolas, which can open upwards or downwards depending on the sign of \( A \). A positive \( A \) generates a parabola opening upwards, while a negative \( A \) opens downwards. These expressions are fundamental as they describe motion, areas, and more.
  • The vertex of the parabola is a key feature and can be found using \(-\frac{B}{2A}\) for its x-coordinate.
  • The y-intercept is easily identifiable as \( C \), the constant term.
Quadratic expressions exhibit diverse behaviors through their roots, extrema, and concavity, significantly impacting mathematical modeling and problem-solving. In our original problem, ensuring all real values of \( x \) yield a non-negative result highlights how crucial evaluating a quadratic's discriminant is in guaranteeing its behavior across the real number line.

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Most popular questions from this chapter

The quadratic equations \(x^{2}-6 x+a=0\) and \(x^{2}-c x+\) \(6=0\) have one root in common. The other roots of the first and second equations are integers in the ratio \(4: 3 .\) Then the common root is (A) 1 (B) 4 (C) 3 (D) 2

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the equation \(x^{2}+2(k+1) x+9 k-5=0\) has only negative roots, then \(k \leq 6\) Reason: The equation \(f(x)=0\) will have both roots negative if and only if (i) Discriminant \(\geq 0\), (ii) Sum of roots \(<0\), (iii) Product of roots \(>0\)

If the ratio of the roots of \(\lambda x^{2}+\mu x+v=0\) is equal to the ratio of the roots of \(x^{2}+x+1=0\), then \(\lambda, \mu, v\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

If \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) are the roots of the equation \(x^{n}+a x+\) \(b=0\), then the value of \(\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)\left(x_{1}-x_{4}\right) \ldots\) \(\left(x_{1}-x_{n}\right)\) is equal to (A) \(n x_{1}^{n-1}+a\) (B) \(n\left(x_{1}\right)^{n-1}\) (C) \(n x_{1}+b\) (D) \(n x_{1}^{n-1}+b\)

Let \(k\) be any point such that \(k \in R\) and \(\alpha, \beta\) are the roots of the quadratic equation \(f(x)=a x^{2}+b x+c=0\). If \(k\) lies outside and is less than both the roots then the equation must have real and distinct roots and the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is less than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If \(k\) lies between both the roots, then the sign of \(f(k)\) is opposite to the sign of ' \(a\) '. If \(k\) lies outside and is greater than both the roots, then the sign of \(f(k)\) is same as the sign of ' \(a\) '. Also, \(k\) is greater than the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\). If both the roots of the equation lie between two real numbers \(k_{1}\) and \(k_{2}\), then equation must have real and distinct roots and the sign of \(f\left(k_{1}\right)\) and \(f\left(k_{2}\right)\) is same as the sign of \(a\). Also, the \(x\)-coordinate of the vertex of the parabola \(y=a x^{2}+b x+c\) lies between \(k_{1}\) and \(k_{2}\). The values of ' \(a\) ' for which the roots of the equation \((a+1) x^{2}-3 a x+4 a=0(a \neq-1)\) to be greater than unity are (A) \(\frac{-16}{7} \leq a<-1\) (B) \(-2
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