Bot GPT 4-o mini To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, first, we need to move the constant term to the other side:
\[ x^2 + 54x + 729 = 0 \]
Next, we can rewrite the left side as a perfect square trinomial. We look for a number that when squared gives us the constant term, which is 729 when we complete the square.
- First, take half of the \( b \) term (which is 54), square it, and check:
\[ \left(\frac{54}{2}\right)^2 = 27^2 = 729 \]
- Now, we can write the left side as a square:
\[ (x + 27)^2 = 729 \]
- Next, we take the square root of both sides:
\[ x + 27 = \pm 27 \]
This gives us two equations to solve:
- \( x + 27 = 27 \)
- \( x + 27 = -27 \)
Solving the first equation:
\[ x + 27 = 27 \ x = 27 - 27 \ x = 0 \]
Solving the second equation:
\[ x + 27 = -27 \ x = -27 - 27 \ x = -54 \]
Thus, the solutions for the equation \( x^2 + 54x = -729 \) are:
\[ x = 0 \quad \text{and} \quad x = -54 \]
Final answer:
\[ x = 0 \text{ or } x = -54 \]
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