To solve the equation [tex]\( x^2 - 3x - 54 = (x + k)(x + m) \)[/tex] and find the value of [tex]\( |k - m| \)[/tex], we follow the steps below:
1. Compare Polynomial Forms:
- First, we expand the factored form to match the standard quadratic polynomial form.
- [tex]\[ (x + k)(x + m) = x^2 + (k + m)x + km \][/tex]
2. Match Coefficients:
- From the given quadratic equation [tex]\( x^2 - 3x - 54 \)[/tex], we note the coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] term is 1.
- The coefficient of [tex]\( x \)[/tex] term is -3.
- The constant term is -54.
- We now compare these coefficients with those from the expanded form:
- Comparing [tex]\( x \)[/tex] term coefficients: [tex]\( k + m = -3 \)[/tex]
- Comparing constant terms: [tex]\( km = -54 \)[/tex]
3. Solve the System of Equations:
- We have the system:
- [tex]\( k + m = -3 \)[/tex]
- [tex]\( km = -54 \)[/tex]
4. Find the Roots [tex]\( k \)[/tex] and [tex]\( m \)[/tex]:
- To find [tex]\( k \)[/tex] and [tex]\( m \)[/tex], we use the system of equations:
- Rewrite the quadratic equation in [tex]\( k \)[/tex] and [tex]\( m \)[/tex]: [tex]\( t^2 + 3t - 54 = 0 \)[/tex]
- Solve for [tex]\( t \)[/tex] using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -54 \)[/tex]:
- [tex]\( t = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-54)}}{2 \cdot 1} \)[/tex]
- [tex]\( t = \frac{-3 \pm \sqrt{9 + 216}}{2} \)[/tex]
- [tex]\( t = \frac{-3 \pm \sqrt{225}}{2} \)[/tex]
- [tex]\( t = \frac{-3 \pm 15}{2} \)[/tex]
- This gives us two solutions: [tex]\( t = \frac{12}{2} = 6 \)[/tex] and [tex]\( t = \frac{-18}{2} = -9 \)[/tex]
- Thus, [tex]\( k = -9 \)[/tex] and [tex]\( m = 6 \)[/tex]
5. Calculate [tex]\(|k - m|\)[/tex]:
- Now we find [tex]\( |k - m| \)[/tex]:
- [tex]\[ |k - m| = |-9 - 6| = |-15| = 15 \][/tex]
Therefore, the value of [tex]\( |k - m| \)[/tex] is [tex]\( \boxed{15} \)[/tex].
