We have mentioned before that the
roots of a quadratic equation are the solutions or answers you get from solving the quadatic equation. Working back from the answers, will take you to an equation.
Find an equation with roots 13 and -5
The step before giving the solutions would be:
Notice that the signs in the brackets are opposite of the given roots.
Of course, there would be other possibilities as well when each term on each side of the
equal to sign is multiplied by a constant.
Find an equation with roots
and 4
Notice that if
then
Therefore the two brackets will be:
The equation is:
Theory of quadratic equations - advanced
This section is not in the syllabus, but it gives one a good understanding about some of the solutions of the quadratic equations.
What is the discriminant of a quadratic equation?
Consider a general quadratic function of the form
. The
discriminant is defined as:
This is the expression under the square root in the formula for the roots of this function. We have already seen that whether the roots exist or not depends on whether this factor
is negative or positive.
The nature of the roots
Real roots (
)
Consider
for some quadratic function
. In this case there are solutions to the equation
given
by the formula
If the expression under the square root is non-negative then the square root exists. These are the roots of the function
.
There various possibilities are summarised in the figure below.
Equal roots (
)
If
, then the roots are equal and, from the formula, these
are given by
Unequal roots (
)
There will be 2 unequal roots if
. The roots of
are
rational if
is a perfect square (a number which is the square of a rational number), since, in this case,
is rational. Otherwise, if
is not a perfect square, then the roots are
irrational .
Imaginary roots (
)
If
, then the solution to
contains the square root of a negative number and therefore there are no real solutions. We therefore say that the roots of
are
imaginary (the graph of the function
does not intersect the
-axis).
Theory of quadratics - advanced exercises
From past papers
[IEB, Nov. 2001, HG] Given:
Show that the discriminant is given by:
If
, discuss the nature of the roots of the equation.
If
, find the value(s) of
for which the roots are equal.
[IEB, Nov. 2002, HG] Show that
has non-real roots for all real values for
.
[IEB, Nov. 2003, HG] The equation
has real roots.
Find the largest integral value of
.
Find one rational value of
, for which the above equation has rational roots.
[IEB, Nov. 2003, HG] In the quadratic equation
,
,
and
are positive real numbers and form a geometric sequence. Discuss the nature of the roots.
[IEB, Nov. 2004, HG] Consider the equation:
Find a value of
for which the roots are equal.
Find an integer
for which the roots of the equation will be rational and unequal.
[IEB, Nov. 2005, HG]
Prove that the roots of the equation
are real for all real values of
,
and
.
When will the roots of the equation be equal?
[IEB, Nov. 2005, HG] If
and
can take on only the values 1, 2 or 3, determine all pairs (
) such that
has real roots.
End of chapter exercises
Solve:
(Give your answer correct to two decimal places.)
Solve:
Solve:
(Hint: Let
and solve for
first and use the answer to solve
.)
Solve for
:
Solve for
:
(Show your answer correct to ONE decimal place.)
(correct to 2 decimal places)
Solve for
by completing the square:
The equation
has roots
and
. Find one set of possible values for
,
and
.
The two roots of the equation
differ by 5. Calculate the value of
.
An equation of the form
is written
on the board. Saskia and Sven copy it down incorrectly. Saskia hasa mistake in the constant term and obtains the solutions -4 and 2.
Sven has a mistake in the coefficient of
and obtains the solutions
1 and -15. Determine the correct equation that was on theboard.
Bjorn stumbled across the following formula to solve
the quadratic equation
in a foreign textbook.
Use this formula to solve the equation:
Solve the equation again, using factorisation, to see if the formula works for this equation.
Trying to derive this formula to prove that it always works, Bjorn got stuck along the way. His attempt his shown
below:
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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