Question
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Answer
The solutions are \( t = 7 + 2\sqrt{3} \) and \( t = 7 - 2\sqrt{3} \).
Solution
Start with the equation:
4(t – 7)² = 48
Step 1: Divide both sides by 4:
(t – 7)² = 12
Step 2: Take the square root of both sides, remembering both positive and negative roots:
t – 7 = ±√12
Step 3: Simplify √12:
√12 = 2√3 (since 12 = 4 × 3 and √4 = 2)
Now, we have:
t – 7 = ±2√3
Step 4: Solve for t by adding 7 to both sides:
t = 7 ± 2√3
So, the solutions are:
t = 7 + 2√3 and t = 7 – 2√3
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Simplify this solution Extra Insights
To start, divide both sides of the equation by 4: \( (t-7)^2 = 12 \). Next, take the square root of both sides: \( t - 7 = \pm \sqrt{12} \). Simplifying \( \sqrt{12} \) gives \( \sqrt{4 \times 3} = 2\sqrt{3} \), so we have: \( t - 7 = \pm 2\sqrt{3} \). Finally, add 7 to both sides to solve for \( t \): \( t = 7 \pm 2\sqrt{3} \). Now you have two solutions! Remember: \( t \) can be either \( 7 + 2\sqrt{3} \) or \( 7 - 2\sqrt{3} \).
