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Chapter 3: Problem 190

\(a, b, \in R, a \neq b\) roots of equation \((a-b) x^{2}+5(a+b) x-2\) \((a-b)=0\) are (a) Real and distinct (b) Complex (c) real and equal (d) None

Short Answer

Expert verified
The roots of the equation \((a-b)x^2 + 5(a+b)x - 2 = 0\) when \((a-b) = 0\) are real and distinct.

Step by step solution

01

Calculate the discriminant

To determine the nature of the roots of the equation, we will compute the discriminant \(D\), which is given by the formula \(D = B^2 - 4AC\), where \(A\), \(B\), and \(C\) are the coefficients of the quadratic equation. In this case, the equation is \((a-b)x^2 + 5(a+b)x - 2 = 0\), so we have: \(A = (a-b)\), \(B = 5(a+b)\), and \(C = -2\). We are given the condition \((a-b) = 0\), which means that \(A = 0\). Now, calculate the discriminant: \(D = [5(a+b)]^2 - 4 \cdot 0 \cdot (-2)\) Since \(A = 0\), we can simplify to: \(D = [5(a+b)]^2\)
02

Determine the nature of the roots

To determine the nature of the roots, we will analyze the value of the discriminant: 1. If \(D > 0\), the roots are real and distinct. 2. If \(D = 0\), the roots are real and equal. 3. If \(D < 0\), the roots are complex. Now, since \(a\neq b\) and \(a, b \in R\) (i.e., both \(a\) and \(b\) are real numbers): \(D = [5(a+b)]^2\) Since squaring any real number will always produce a non-negative result, we can say that \(D \ge 0\). Since \(a\neq b\), the expression \((a+b)\) must be non-zero and therefore, \(D > 0\). So, the correct option is: (a) Real and distinct

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Most popular questions from this chapter

If \(\alpha, \beta\) are roots of \(x^{2}+p x+q=0\) and \(x^{2 n}+p^{n} x^{n}+q^{n}=0\) and if \((\alpha / \beta)\) is one root of \(\mathrm{x}^{\mathrm{n}}+1+(\mathrm{x}+1)^{\mathrm{n}}=0\) then \(\mathrm{n}\) must be (a) even integer (b) odd integer (c) rational but not integer (d) None of these

The sum of the roots of the equation \(\mathrm{x}^{2}-3|\mathrm{x}|-10=0\) is (a) 3 (b) \(-3\) (c) \(-10\) (d) 0

Discriminant of the quadratic equation \(\sqrt{5} \mathrm{x}^{2}-3 \sqrt{3} \mathrm{x}-2 \sqrt{5}=0\) is (a) 67 (b) 76 (c) \(-67\) (d) \(-76\)

The number of real values of \(\mathrm{x}\) satisfying the equation \(3\left[x^{2}+\left(1 / x^{2}\right)\right]-16[x+(1 / x)]+26=0\) is (a) 1 (b) 2 (c) 3 (d) 4

If the difference of the roots of the equation \(x^{2}-p x+q=0\) is 1 then (a) \(p^{2}+4 q^{2}=(1+2 q)^{2}\) (b) \(\mathrm{q}^{2}+4 \mathrm{p}^{2}=(1+2 \mathrm{q})^{2}\) (c) \(p^{2}-4 q^{2}=(1+2 q)^{2}\) (d) \(q^{2}+4 p^{2}=(1-2 q)^{2}\)
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