To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first need to move all terms to one side of the equation:
\[ 4x^2 + 12x + 5 + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 12x + 9 = 0 \]
Now, we can factor the quadratic equation. We are looking for two numbers that multiply to \( 4 \cdot 9 = 36 \) and add up to \( 12 \). The numbers \( 6 \) and \( 6 \) meet these criteria.
Thus, we can rewrite the middle term:
\[ 4x^2 + 6x + 6x + 9 = 0 \]
Next, we can factor by grouping:
\[ (4x^2 + 6x) + (6x + 9) = 0 \]
Factoring out the common terms in each group:
\[ 2x(2x + 3) + 3(2x + 3) = 0 \]
Now we can factor out the common binomial factor \( (2x + 3) \):
\[ (2x + 3)(2x + 3) = 0 \]
Or simply:
\[ (2x + 3)^2 = 0 \]
Setting the factor equal to zero gives us:
\[ 2x + 3 = 0 \]
Solving for \( x \):
\[ 2x = -3 \quad \Rightarrow \quad x = -\frac{3}{2} \]
Thus, the solution to the equation \( 4x^2 + 12x + 5 = -4 \) is:
\[ x = -\frac{3}{2} \]
1 answer