To solve the equation
[tex]\[
\frac{10 x + 25}{3 x + 12} = \frac{5 x}{x + 4},
\][/tex]
we will follow these steps:
1. Understand that we want to find [tex]\( x \)[/tex] values that satisfy this equality.
2. Cross-multiply to clear the denominators:
[tex]\[
(10 x + 25)(x + 4) = (5 x)(3 x + 12).
\][/tex]
3. Expand both sides of the equation:
[tex]\[
(10 x + 25)(x + 4) = 10x^2 + 40x + 25x + 100 = 10x^2 + 65x + 100,
\][/tex]
[tex]\[
(5 x)(3 x + 12) = 15x^2 + 60x.
\][/tex]
4. Equate the expanded forms:
[tex]\[
10x^2 + 65x + 100 = 15x^2 + 60x.
\][/tex]
5. Move all terms to one side to form a standard quadratic equation:
[tex]\[
10x^2 + 65x + 100 - 15x^2 - 60x = 0,
\][/tex]
[tex]\[
-5x^2 + 5x + 100 = 0.
\][/tex]
6. Simplify the quadratic equation:
[tex]\[
x^2 - x - 20 = 0.
\][/tex]
7. Solve this quadratic equation using any appropriate method (factoring, completing the square, or the quadratic formula).
After solving, we find valid values for [tex]\( x \)[/tex]. These solutions must be checked against the original equation to confirm they do not create denominators of zero.
The result from our solution process shows that the equation has
[tex]\[
1
\][/tex]
valid solution.
