Question

If the sum of the two roots of the equation \([1 /(x+a)]+[1 /(x+b)]=(1 / k)\) is zero then their Product is (a) \(\overline{(1 / 2)\left(a^{2}+b^{2}\right)}\) (b) \(\\{(-1) / 2\\}(a+b)^{2}\) (c) \([\\{a+b\\} / 2]^{2}\) (d) None

Short answer

The product of the roots is \(\frac{(-1)(ab)}{4}\), which does not match any of the given answer choices. Therefore, the correct answer is (d) None.

Step-by-step solution

Rewrite the equation in a more simplified form

We have to rewrite the given equation in a more simplified form to be able to deal with it. The given equation is: \( \frac{1}{x+a} + \frac{1}{x+b} = \frac{1}{k} \) Let's get Common Denominator and then simplify: \(\frac{x+b}{(x+a)(x+b)} + \frac{x+a}{(x+a)(x+b)} = \frac{1}{k}\)

Rewrite the equation in quadratic form

Now, sum the numerators: \(\frac{2x+a+b}{(x+a)(x+b)} = \frac{1}{k}\) Multiply both sides by \(k(x+a)(x+b)\): \(2xk(x+a)(x+b) + (a+b)k(x+a)(x+b) = (x+a)(x+b)\) Now, let's rearrange and group terms to rewrite it as a quadratic equation. We will have: \(x^2(2k - 1) + x((2a+2b)k - a - b) + abk = 0\)

Apply the condition (sum of the roots is zero)

We are given that the sum of the roots is zero. The sum of the roots of a quadratic equation \(Ax^2 + Bx + C = 0\) can be given by \(-\frac{B}{A}\). So: \(-\frac{(2a+2b)k - a - b}{2k - 1} = 0\) Since \(2k - 1 ≠ 0\) : \((2a+2b)k = a + b\) Divided by \((a+b)\): \(2k = \frac{a + b}{a + b}\) \[2k = 1\] Hence, we find: \(k = \frac{1}{2}\)

Calculate the product of the roots

The product of the roots of a quadratic equation \(Ax^2 + Bx + C = 0\) can be given by \(\frac{C}{A}\). So: \(\frac{abk}{2k-1} = \frac{ab(\frac{1}{2})}{\frac{1}{2} - 1}\) \(\frac{ab}{2}\left(\frac{1}{2}\right)\left(\dfrac{-1}{\frac{1}{2}}\right) = \frac{(-1)(ab)}{4}\) The product of the roots is \(\frac{(-1)(ab)}{4}\). This does not match any of the given answer choices, so the correct answer is: (d) None

Key concepts

Roots of Quadratic Equation

When we talk about quadratic equations, we're looking at polynomial equations of the second degree, which can be generally represented as \( Ax^2 + Bx + C = 0 \). Here, \( A \), \( B \), and \( C \) are coefficients, with \( A \) not being zero, since otherwise, the equation would not be quadratic. The solutions to this equation are termed as the 'roots' or 'zeros' of the quadratic equation.

Roots can be real or complex numbers and they can be distinct or the same (repeated roots). To find the roots of a quadratic equation, we can use various methods such as factoring, completing the square, using the quadratic formula, or graphically analyzing the function \( y = Ax^2 + Bx + C \).

The roots are especially significant in many applications of mathematics and science, such as physics, where they can represent time at which certain events occur or in economics to determine levels of production that maximize profit, to give a few examples.

Sum and Product of Roots

The relationship between the roots of a quadratic equation and its coefficients is profoundly expressed by two important formulas. For a quadratic equation \( Ax^2 + Bx + C = 0 \), if the roots are \( \text{root}_1 \) and \( \text{root}_2 \), then their sum and product can be given by the following:

• Sum of the roots: \( \text{root}_1 + \text{root}_2 = -\frac{B}{A} \)
• Product of the roots: \( \text{root}_1 \times \text{root}_2 = \frac{C}{A} \)

These formulas come in handy in solving problems without actually finding the roots themselves, as seen in the given exercise. By understanding the relationship between the coefficients and the roots, one can often determine certain properties of the roots (like their sum or product) quickly and efficiently.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves rewriting expressions in a more compact or simplified form, without changing their value. This process could involve combining like terms, using the distributive property, and cancelling out terms where possible.

For instance, consider combining like terms. If one has an expression like \( 3x + 2x \), this simplifies to \( 5x \) because both terms are 'like' terms (terms that have the same variables raised to the same power). The distributive property might also be used to simplify expressions such as \( 3(2x + 4) \), which simplifies to \( 6x + 12 \).

The step-by-step solution provided integrates these concepts as it breaks down the equation into a quadratic form and simplifies it to find the sum and product of its roots. The goal when simplifying an algebraic expression is ultimately to make the equation or expression clearer and more manageable for solving or understanding the relations within it.