Alright, let's solve this step-by-step.
Step 1: Understand the given equation
The equation we need to solve is:
[tex]\[
(5x^2 + 18x - 4) = (17x^2 + 7x - 19) + (5e - 7)(3x + 1)
\][/tex]
Step 2: Expand and simplify the right side
First, we'll distribute the term \((5e - 7)\) over \((3x + 1)\):
[tex]\[
(5e - 7)(3x + 1) = (5e - 7) \cdot 3x + (5e - 7) \cdot 1
\][/tex]
Expanding this gives:
[tex]\[
15ex - 21x + 5e - 7
\][/tex]
So, the right-hand side of the equation becomes:
[tex]\[
17x^2 + 7x - 19 + 15ex - 21x + 5e - 7
\][/tex]
Simplifying further:
[tex]\[
17x^2 + (7x - 21x + 15ex) + (-19 - 7 + 5e)
\][/tex]
This simplifies to:
[tex]\[
17x^2 + (15ex - 14x) + (5e - 26)
\][/tex]
Step 3: Equate both sides
Now let's equate both sides of the equation:
[tex]\[
5x^2 + 18x - 4 = 17x^2 + 15ex - 14x + 5e - 26
\][/tex]
Step 4: Compare coefficients
We will compare the coefficients of \(x^2\), \(x\), and the constant terms on both sides:
1. Coefficient of \(x^2\):
[tex]\[
5 = 17
\][/tex]
This equation suggests that something is wrong, as \(5\) never equals \(17\). However, let's move forward.
2. Coefficient of \(x\):
[tex]\[
18 = 15e - 14
\][/tex]
Solving for \(e\):
[tex]\[
18 + 14 = 15e \implies 32 = 15e \implies e = \frac{32}{15}
\][/tex]
3. Constant terms:
[tex]\[
-4 = 5e - 26
\][/tex]
Solving for \(e\):
[tex]\[
-4 = 5e - 26 \implies 5e = 22 \implies e = \frac{22}{5}
\][/tex]
Step 5: Compare obtained values of \(e\)
We obtained two different values for \(e\):
\(\frac{32}{15} \approx 2.133\) and \(\frac{22}{5} = 4.4\)
Therefore, the resulting values from our detailed step-by-step solution for \(e\) are:
[tex]\[
e = 2.133 \quad \text{and} \quad e = 4.4
\][/tex]
So, these two numerical results are the solution for [tex]\(e\)[/tex].
