Although these solutions were carefully checked, the solver is only human. Kindly comment in the box below if you see any errors.
31.) Factor completely: .
Rearranging the terms, we obtain
Factoring by grouping, we have
Answer: (a + b)(a – b)(c + d)(c – d)
32.) Simplify .
33.) Perform the indicated operations:
34.) Perform the indicated operations.
We can factor the given completely into the following expressions and change division into multiplication by multiplying the first two expressions with the reciprocal of the third expression. The result of these operations is shown below.
After cancelling similar terms, we are left with
35.) Simplify .
Simplifying the rational expressions, we have
This simplifies further to
36.) Solve for x in the equation
Getting the least common denominator of the left hand side and combining the terms, we have
Now, we can only equation the numerator.
which is equivalent to $x^2 – 3x + 2 = 0$
Getting the solution, we have
Therefore, or . But x cannot be equal to 2, because the it will make the denominator of the original expressions undefined. So, the solution is .
Answer: x = 1
37.) The points (0, 0), (2, 3), and (4, 0) form a triangle. What is its perimeter?
To find the perimeter of the triangle, we need to find the distance between these points and add them. Let’s name the points A, B, and C respectively.
Using the distance formula, we can subtract the corresponding coordinates, square them, and get the square root.
Distance between A and B is
Distance between A and C is
Distnace between B and C is
So, the perimeter of ABC is
38.) Find the equation of the perpendicular bisector of the segment joining the points (-3, 2) and (5, 2)?
Let A be the point with coordinates (-3,2) and B be the point with coordinates (5,2). Notice that AB have the same y-coordinates which mean that it is a horizontal segment. This means that the perpendicular bisector is a vertical line.
To get the perpendicular bisector, we get the midpoint M of AB and find the equation of the vertical line passing through M.
So, the equation of the perpendicular bisector of AB is
Answer: x = 1
39.) A man agrees to invest part of his 1-million-peso inheritance at an annual interest rate of 5%, while the rest at 6% interest. If, at the end of the year, he needs a total interest of Php 56, 200, how much should he invest at 5%?
Let x = amount invested at 5% and 1000 000 – x be invested at 6%.
40.) If 2x + 5y = 10 and x = 3y + 1, what is 11x + 11y?
Substituting the expression on the right hand side of the second equation to x in the left hand side of the first equation, we have
Multiplying by 3, we get (#).
Transposing the second equation and multiplying it by 5, we have
Adding # and ##, we have
By * and **
We have 11x + 11y = 35 + 8 = 43$.
We will discuss the solution of numbers 41-50 in the next post.