Sure, let's solve the problem step by step.
Given the quadratic equation:
[tex]\[ x^2 - 3x - 54 = (x + k)(x + m) \][/tex]
We need to find the values of [tex]\( k \)[/tex] and [tex]\( m \)[/tex] and then compute [tex]\( |k - m| \)[/tex].
### Step 1: Expand the right-hand side of the equation
If we expand the right-hand side, we get:
[tex]\[ (x + k)(x + m) = x^2 + (k + m)x + km \][/tex]
### Step 2: Compare the expanded form with the original equation
When we compare it with the original equation [tex]\( x^2 - 3x - 54 \)[/tex], we obtain two pieces of information:
1. The coefficient of [tex]\( x \)[/tex] on the left-hand side must equal the coefficient of [tex]\( x \)[/tex] on the right-hand side.
2. The constant term on the left-hand side must equal the constant term on the right-hand side.
From the comparison, we have:
[tex]\[ k + m = -3 \][/tex]
[tex]\[ km = -54 \][/tex]
### Step 3: Solve the system of equations to find the values of [tex]\( k \)[/tex] and [tex]\( m \)[/tex]
We now have a system of equations:
1. [tex]\( k + m = -3 \)[/tex]
2. [tex]\( km = -54 \)[/tex]
One way to find the values of [tex]\( k \)[/tex] and [tex]\( m \)[/tex] is to solve the quadratic equation whose roots are [tex]\( k \)[/tex] and [tex]\( m \)[/tex]:
[tex]\[ x^2 + (k + m)x + km = 0 \][/tex]
Using the given values [tex]\( k + m = -3 \)[/tex] and [tex]\( km = -54 \)[/tex]:
[tex]\[ x^2 - 3x - 54 = 0 \][/tex]
This is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -54 \)[/tex].
### Step 4: Use the quadratic formula to find the roots
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -54 \)[/tex]:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 + 216}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{225}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm 15}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{3 + 15}{2} = \frac{18}{2} = 9 \][/tex]
[tex]\[ x = \frac{3 - 15}{2} = \frac{-12}{2} = -6 \][/tex]
So, [tex]\( k \)[/tex] and [tex]\( m \)[/tex] are 9 and -6 (in any order).
### Step 5: Compute [tex]\( |k - m| \)[/tex]
[tex]\[ |k - m| = |9 - (-6)| \][/tex]
[tex]\[ |k - m| = |9 + 6| \][/tex]
[tex]\[ |k - m| = |15| \][/tex]
[tex]\[ |k - m| = 15 \][/tex]
### Final Answer:
The value of [tex]\( |k - m| \)[/tex] is:
[tex]\[ 15 \][/tex]
