# Tutor profile: Madison Z.

## Questions

### Subject: Pre-Calculus

Find the vertex of the parabola $(y = 2x^2 + 3x + 4.$)

The x-coordinate of the vertex of a parabola is given by the equation $(x_{vertex} = -\frac{b}{2a},$) where a and b are defined by the standard form of a quadratic equation: $(y = ax^2 + bx + c.$) In this example, a=2 and b=3. We can plug these into the vertex equation: $(x_{vertex} = -\frac{3}{2(2)} = -\frac{3}{4}.$) To find the y-coordinate of the vertex, we can substitute the x-coordinate we solved for into the original equation. $(y = 2(-\frac{3}{4})^2 + 3(-\frac{3}{4}) + 4$) $(y = 2(\frac{9}{16}) + 3(-\frac{3}{4}) + 4$) $(y = \frac{18}{16} - \frac{9}{4} + 4$) $(y = \frac{18}{16} - \frac{36}{16} + \frac{64}{16}$) $(y = \frac{18-36+64}{16}= \frac{46}{16} = \frac{23}{8}$) Therefore, the vertex is located at $((-\frac{3}{4}, \frac{23}{8}).$)

### Subject: Pre-Algebra

$(\frac{2}{3} + \frac{4}{5} = ?$)

Here we are adding two fractions with unlike denominators. First, we need to find a common denominator. We can do so by finding the least common multiple, or LCM, of 3 and 5. Multiples of 3 - 3, 6, 9, 12, 15 Multiples of 5 - 5, 10, 15, 20, 25 The LCM of 3 and 5 is 15. Now we need to rewrite each fraction with the new denominator. We know that we multiply the denominator, 3, by 5 to get 15. So we need to find an equivalent fraction by multiplying the numerator and denominator by 5. $(\frac{2*5}{3*5} = \frac{10}{15}$) We do the same to the second fraction, only now we multiply by 3. $(\frac{4*3}{5*3} = \frac{12}{15}$). Now our problem looks like $(\frac{10}{15} + \frac{12}{15} = ?$) Now that we have a common denominator, now we can add the numerators together. $(\frac{10}{15} + \frac{12}{15} = \frac{22}{15}$) We cannot reduce this fraction, but we can rewrite as a mixed number. How many whole 15s can fit into 22? Just 1. How much is left over? 7 (22-15=7). So, our final answer is $(\frac{2}{3} + \frac{4}{5} = 1 \frac{7}{15}$)

### Subject: Basic Math

Susan and Lindsey share a gardening business. They made $240 this week. If Susan worked 7 hours this week and Lindsey worked 5 hours, how should they divide their earnings fairly?

This is an example of a proportional reasoning problem. If Susan worked 7 hours and Lindsey worked 5, then they worked a total of 12 hours (7+5=12). If they earned $240 and worked 12 hours, then the hourly rate is $20 (240/12=20). Now we can distribute the money fairly. Susan earned $140 (7*20=140) and Lindsey earned $100 (5*20=100).

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