If \( x = 3 \), then \( x - 1 = 2 > 0 \), valid.
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- So, \( x^2 - 1 = 2^3 = 8 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3 \).
- Check domain: arguments of logs must be positive.
- If \( x = -3 \), then \( x - 1 = -4 < 0 \), invalid.
- Thus, the only solution is \( x = 3 \).
- A quadratic function passes through the points (1, 4), (2, 3), and (4, 15). What is the y-coefficient of the x-term in its standard form?
- Let the quadratic be \( f(x) = ax^2 + bx + c \).
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Solving these equations simultaneously:
The cosine function is positive in Quadrants I and IV, yielding two solutions per period:
Vectors are perpendicular if their dot product is zero:
Let the triangle be a right triangle with legs $ a $ and $ b $, and hypotenuse $ c $. The area $ A $ of the triangle is given by:
The radius $ r $ of the inscribed circle in a right triangle is:
Substitute into the formula for $ r $:
Now, recall that the area can also be expressed in terms of the inradius and semiperimeter $ s $:
Question: A triangle has sides of lengths $ 13 $, $ 14 $, and $ 15 $. What is the length of the shortest altitude?
Let the triangle have sides $ a = 13 $, $ b = 14 $, $ c = 15 $. The shortest altitude corresponds to the longest side, which is $ c = 15 $.
First, compute the area using Herons formula.