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Magazine Chapter 0: Problem 102
Find \(k\) if \((x-k)(x+3 k)=x^{2}+4 x-12\).
Short Answer
Expert verified
k = 2
Step by step solution
01
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).
02
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: \(-k \cdot 3k = -3k^2\)
03
Combine Like Terms
Add the results from the FOIL method to get: \(x^2 + 3kx - kx - 3k^2 = x^2 + 2kx - 3k^2\)
04
Equate the Expanded Form to the Right-Hand Side
Set the expanded expression equal to the right-hand side of the original equation: \(x^2 + 2kx - 3k^2 = x^2 + 4x - 12\)
05
Compare Coefficients
Compare the coefficients of like terms on both sides of the equation. For the linear term: \(2k = 4\). For the constant term: \(-3k^2 = -12\).
06
Solve for \(k\)
Solve the linear coefficient equation first: \(2k = 4\) \(k = 2\). Verify with the constant term equation: \(-3k^2 = -12\) \(-3(2)^2 = -12\) \(-12 = -12\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
\( x^2 + 3kx - kx - 3k^2 \)
- **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:
\[ x^2 + 3kx - kx - 3k^2 \]
\[ (x - k)(x + 3k) = x(x + 3k) - k(x + 3k) \]
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 \( k \).
\[ x^2 + 2kx - 3k^2 = x^2 + 4x - 12 \]
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.
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.
