Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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]
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
