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Consider this equation.
[tex]
\frac{10x + 25}{3x + 12} = \frac{5x}{x + 4}
[/tex]

How many valid solutions does the equation have?

The equation has [tex]\square[/tex] valid solution(s).


Sagot :

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.
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