Question

Find \(k\) if \((x-k)(x+3 k)=x^{2}+4 x-12\).

Short answer

k = 2

Step-by-step solution

Expand the Left-Hand Side

To find the value of \(k\), expand the expression on the left-hand side of the equation \((x - k)(x + 3k)\). Use the distributive property (also known as FOIL for binomials).

Apply FOIL to Expand

Multiply the first, outer, inner, and last terms in the binomials: \((x - k)(x + 3k)\) First: \(x \cdot x = x^2\) Outer: \(x \cdot 3k = 3kx\) Inner: \(-k \cdot x = -kx\) Last: \(-k \cdot 3k = -3k^2\)

Combine Like Terms

Add the results from the FOIL method to get: \(x^2 + 3kx - kx - 3k^2 = x^2 + 2kx - 3k^2\)

Equate the Expanded Form to the Right-Hand Side

Set the expanded expression equal to the right-hand side of the original equation: \(x^2 + 2kx - 3k^2 = x^2 + 4x - 12\)

Compare Coefficients

Compare the coefficients of like terms on both sides of the equation. For the linear term: \(2k = 4\). For the constant term: \(-3k^2 = -12\).

Solve for \(k\)

Solve the linear coefficient equation first: \(2k = 4\) \(k = 2\). Verify with the constant term equation: \(-3k^2 = -12\) \(-3(2)^2 = -12\) \(-12 = -12\).

Key concepts

FOIL method

When dealing with the multiplication of two binomials, a helpful shortcut to remember is the FOIL method. FOIL stands for First, Outer, Inner, Last. Each term represents a specific multiplication step:

- **First**: Multiply the first terms in each binomial.
- **Outer**: Multiply the outer terms in the binomials.
- **Inner**: Multiply the inner terms in the binomials.
- **Last**: Multiply the last terms in each binomial.

Using the FOIL method for \texttt{(x - k)(x + 3k)} gives us:

- **First**: \( x \times x = x^2 \)
- **Outer**: \( x \times 3k = 3kx \)
- **Inner**: \( -k \times x = -kx \)
- **Last**: \( -k \times 3k = -3k^2 \)

Combining all these, we get:

\( x^2 + 3kx - kx - 3k^2 \)

Distributive property

The distributive property is a fundamental algebraic property that allows you to multiply a sum by multiplying each addend separately and then add the products. When expanding \texttt{(x - k)(x + 3k)} using the distributive property, you distribute each term in the first binomial to every term in the second binomial:

\[ (x - k)(x + 3k) = x(x + 3k) - k(x + 3k) \]

Breaking this down, we get:

- \(x \times x = x^2 \)
- \(x \times 3k = 3kx \)
- \(-k \times x = -kx \)
- \(-k \times 3k = -3k^2 \)

Thus, the expression becomes:

\[ x^2 + 3kx - kx - 3k^2 \]

Comparing coefficients

Once we've expanded and simplified the left-hand side of the equation, we equate it to the right-hand side to compare coefficients. The equation derived is:

\[ x^2 + 2kx - 3k^2 = x^2 + 4x - 12 \]

We compare the coefficients of similar terms on both sides:

- Coefficient of \( x^2 \): Both sides have \( x^2 \), so this is equal.
- Coefficient of \( x \): On the left it's \( 2k \), and on the right, it's \( 4 \). Therefore, \( 2k = 4 \).
- Constant term: On the left, it's \( -3k^2 \) and on the right, it's \( -12 \). Hence, \( -3k^2 = -12 \).

Comparing these coefficients allows us to set up separate equations to solve for \( k \).

Solving for constants

Solving for the constants involves taking the equations set up from comparing coefficients and solving them:

First, we solve \( 2k = 4 \):

\[ 2k = 4 \]

Divide both sides by 2:

\[ k = 2 \]

Next, we verify with the constant term equation \( -3k^2 = -12 \):

Substitute \( k = 2 \):

\[ -3(2)^2 = -12 \]

\[ -3 \times 4 = -12 \]

\[ -12 = -12 \]

This confirms that our value of \( k \) is correct. Therefore, \( k = 2 \) is the solution.