Solve each of the following use quadratic formula
Question 1:
Answer:
Comparing with
, we get
Question 2:
Answer:
Comparing with
, we get
Question 3:
Answer:
Comparing with
, we get
Question 4:
Answer:
Comparing with
, we get
Question 5:
Answer:
Comparing with
, we get
Question 6:
Answer:
Comparing with
, we get
Answer:
Multiplying by 6
or
Comparing with
, we get
Answer:
Multiplying by 15 and simplifying we get
or
Comparing with
, we get
Question 9:
Answer:
or
Comparing with
, we get
Answer:
or
Comparing with
, we get
Answer:
Comparing with
, we get
Question 12:
Answer:
Comparing with
, we get
Answer:
Comparing with
, we get
Answer:
Comparing with
, we get
Without solving, comment on the roots of each of the equations
Notes: The nature of the roots depends on the value of the . Also in the equation
,
are real and
. The following cases can happen:
i) : In this case the roots would be real and equal.
ii) : In this case the roots would be real and unequal.
iii) : In this case the roots would be imaginary numbers
Question 15:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and unequal.
Question 16:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and unequal.
Question 17:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and equal
Question 18:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and unequal
Question 19:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and unequal
Question 20:
Answer:
Comparing with
, we get
Hence : In this case the roots would be real and unequal
Find the value of if the quadratic equations have equal roots:
Question 21:
Answer:
Comparing with
, we get
For roots to be equal, we should have
Substituting
Question 22:
Answer:
Comparing with
, we get
For roots to be equal, we should have
Substituting
Question 23:
Answer:
Comparing with
, we get
For roots to be equal, we should have
or
Question 24:
Answer:
Comparing with
, we get
For roots to be equal, we should have