Write the following as a single fraction in its simplest form - Junior Cycle Mathematics - Question 12 - 2021
Question 12
Write the following as a single fraction in its simplest form.
$$\frac{2}{n - 3} - \frac{5}{2n + 5}$$
Show that $$ (4x - 3)^2 + 24x $$ is positive for all values o... show full transcript
Worked Solution & Example Answer:Write the following as a single fraction in its simplest form - Junior Cycle Mathematics - Question 12 - 2021
Step 1
Write the following as a single fraction in its simplest form.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To combine the two fractions, we will use a common denominator. The common denominator is
(n−3)(2n+5).
Rewrite the fractions:
n−32=(n−3)(2n+5)2(2n+5)
and
2n+55=(n−3)(2n+5)5(n−3).
Combine the fractions:
(n−3)(2n+5)2(2n+5)−5(n−3).
Simplify the numerator:
2(2n+5)−5(n−3)=4n+10−5n+15=−n+25.
Thus, we have:
(n−3)(2n+5)−n+25.
In its simplest form, the answer is:
(n−3)(2n+5)−n+25.
Step 2
Show that (4x - 3)^2 + 24x is positive for all values of x ∈ R.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To demonstrate that (4x−3)2+24x is positive, we first expand the expression:
Expanding the squared term:
(4x−3)2=16x2−24x+9.
Substituting this back into the expression gives:
16x2−24x+9+24x=16x2+9.
Notice that 16x2 is always non-negative (it is a squared term) and adds a positive value, 9:
16x2+9>0 for all x∈R.
Thus, we can conclude that the entire expression is positive for all real values of x.
Join the Junior Cycle students using SimpleStudy...
