Question

Find \(k\) so that $$ \frac{x^{2}+3 k x-3 x-9 k}{x^{2}+2 x-15}=\frac{x+12}{x+5} $$

Short answer

k = 4

Step-by-step solution

- Simplify the Denominator of the Given Fraction

Notice that the denominator of the left-hand side is given as: \(x^{2}+2x-15\). This can be factored into: \((x+5)(x-3)\).

- Identify the Condition for Equality

For the two fractions to be equal, the numerators must also be equal when the denominators are the same.

- Simplify the Numerator of the Left Fraction

Simplify the numerator of the left fraction: \(x^{2} + 3kx - 3x - 9k\). Group the terms as: \((x^{2} - 3x) + (3kx - 9k)\). Factor by grouping: \[x(x - 3) + 3k(x - 3)\]. This results in: \[(x + 3k)(x - 3)\].

- Set Up the Equality of Numerators

Given that the denominators are equal, set the simplified numerators equal to each other: \((x + 3k)(x - 3) = (x + 12)(x - 3)\).

- Cancel Out the Common Factor

Cancel the common factor \((x - 3)\) from both sides to get: \(x + 3k = x + 12\).

- Solve for k

Subtract \(x\) from both sides to isolate \(k\): \(3k = 12\). Divide by 3: \(k = 4\).

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra. When you factor a polynomial, you break it down into simpler components, or factors, that can be multiplied together to get the original polynomial. To illustrate this: in the exercise, the given polynomial in the denominator is \(x^2 + 2x - 15\).

This polynomial needs to be factored so that we can work with simpler expressions. We can factor it into two binomials: \((x + 5)(x - 3)\). By recognizing patterns and using factoring techniques, such as grouping or using special product formulas, you can simplify complex polynomials, making it easier to analyze and solve equations.

Numerator Equality

The concept of numerator equality is vital when dealing with rational expressions. For two fractions to be equal, their numerators also need to be equal given that their denominators are the same.

In our exercise, the left-hand side's numerator is \(x^2 + 3kx - 3x - 9k\). We can simplify this numerator by grouping: \((x^2 - 3x) + (3kx - 9k)\) and then factoring by grouping. This results in \(x(x - 3) + 3k(x - 3)\), which factors further to \((x + 3k)(x - 3)\).

Equating this numerator to the simplified numerator on the right-hand side, \((x + 12)(x - 3)\), allows us to set up an equality: \((x + 3k)(x - 3) = (x + 12)(x - 3)\). By canceling common factors on both sides of the equation, we find that the numerators must be equal for the fractions to be equal.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as simple as possible. It's achieved by combining like terms, factoring, and reducing fractions.

In our problem, we first simplify the denominator and the numerator. After factoring the given expressions, we use logical steps to cancel identical factors in the numerator and the denominator: \((x - 3)(x + 3k)\) in the numerator and \((x - 3)(x + 12)\). We then cancel the common factor \((x - 3)\).

This step helps to get the simpler expression \(x + 3k = x + 12\). From this simplified form, it's straightforward to solve for the variable \(k\) by isolating it: subtract \(x\) from both sides, then divide the remaining equation by 3, arriving at \(k = 4\). Simplifying expressions substantially eases solving complex algebraic problems.