In solving the equation (x + 3)(x + 1) = 48, Eric stated that the solution would be
x + 3 = 48 => x = 45
or
(x + 1) = 48 => x = 47
However, at least one of these solutions fails to work when substituted back into the original equation. Why is that? Please help Eric to understand better; solve the problem yourself, and explain your reasoning.
To solve the equation (x + 3)(x + 1) = 48 correctly, we need to apply the distributive property to expand the equation:
(x + 3)(x + 1) = 48
x(x + 1) + 3(x + 1) = 48
x^2 + x + 3x + 3 = 48
x^2 + 4x + 3 = 48
Now, let's rearrange the equation and set it equal to zero:
x^2 + 4x + 3 - 48 = 0
x^2 + 4x - 45 = 0
We can now factor this quadratic equation to find the values of x:
(x + 9)(x - 5) = 0
Setting each factor equal to zero, we get:
x + 9 = 0 or x - 5 = 0
Solving each equation separately, we find:
x = -9 or x = 5
Now, let's substitute these values back into the original equation to verify if they work:
For x = -9:
(x + 3)(x + 1) = 48
(-9 + 3)(-9 + 1) = 48
(-6)(-8) = 48
48 = 48
For x = 5:
(x + 3)(x + 1) = 48
(5 + 3)(5 + 1) = 48
(8)(6) = 48
48 = 48
Both values of x, -9 and 5, satisfy the original equation.
Therefore, Eric's initial solutions, x = 45 and x = 47, are incorrect. It seems that Eric made a mistake while applying the distributive property or simplifying the equation.