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Finding an equation when you know its roots

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

  1. The step before giving the solutions would be:

    ( x - 13 ) ( x + 5 ) = 0

    Notice that the signs in the brackets are opposite of the given roots.

  2. x 2 - 8 x - 65 = 0

    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 - 3 2 and 4

  1. Notice that if x = - 3 2 then 2 x + 3 = 0

    Therefore the two brackets will be:

    ( 2 x + 3 ) ( x - 4 ) = 0
  2. The equation is:

    2 x 2 - 5 x - 12 = 0

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 f ( x ) = a x 2 + b x + c . The discriminant is defined as:

Δ = b 2 - 4 a c .

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 Δ is negative or positive.

The nature of the roots

Real roots ( Δ 0 )

Consider Δ 0 for some quadratic function f ( x ) = a x 2 + b x + c . In this case there are solutions to the equation f ( x ) = 0 given by the formula

x = - b ± b 2 - 4 a c 2 a = - b ± Δ 2 a

If the expression under the square root is non-negative then the square root exists. These are the roots of the function f ( x ) .

There various possibilities are summarised in the figure below.

Equal roots ( Δ = 0 )

If Δ = 0 , then the roots are equal and, from the formula, these are given by

x = - b 2 a

Unequal roots ( Δ > 0 )

There will be 2 unequal roots if Δ > 0 . The roots of f ( x ) are rational if Δ is a perfect square (a number which is the square of a rational number), since, in this case, Δ is rational. Otherwise, if Δ is not a perfect square, then the roots are irrational .

Imaginary roots ( Δ < 0 )

If Δ < 0 , then the solution to f ( x ) = a x 2 + b x + c = 0 contains the square root of a negative number and therefore there are no real solutions. We therefore say that the roots of f ( x ) are imaginary (the graph of the function f ( x ) does not intersect the x -axis).

Khan academy video on quadratics - 4

Theory of quadratics - advanced exercises

From past papers

  1. [IEB, Nov. 2001, HG] Given:     x 2 + b x - 2 + k ( x 2 + 3 x + 2 ) = 0 ( k - 1 )
    1. Show that the discriminant is given by:
      Δ = k 2 + 6 b k + b 2 + 8
    2. If b = 0 , discuss the nature of the roots of the equation.
    3. If b = 2 , find the value(s) of k for which the roots are equal.
  2. [IEB, Nov. 2002, HG] Show that k 2 x 2 + 2 = k x - x 2 has non-real roots for all real values for k .
  3. [IEB, Nov. 2003, HG] The equation x 2 + 12 x = 3 k x 2 + 2 has real roots.
    1. Find the largest integral value of k .
    2. Find one rational value of k , for which the above equation has rational roots.
  4. [IEB, Nov. 2003, HG] In the quadratic equation p x 2 + q x + r = 0 , p , q and r are positive real numbers and form a geometric sequence. Discuss the nature of the roots.
  5. [IEB, Nov. 2004, HG] Consider the equation:
    k = x 2 - 4 2 x - 5 where x 5 2
    1. Find a value of k for which the roots are equal.
    2. Find an integer k for which the roots of the equation will be rational and unequal.
  6. [IEB, Nov. 2005, HG]
    1. Prove that the roots of the equation x 2 - ( a + b ) x + a b - p 2 = 0 are real for all real values of a , b and p .
    2. When will the roots of the equation be equal?
  7. [IEB, Nov. 2005, HG] If b and c can take on only the values 1, 2 or 3, determine all pairs ( b ; c ) such that x 2 + b x + c = 0 has real roots.

End of chapter exercises

  1. Solve: x 2 - x - 1 = 0    (Give your answer correct to two decimal places.)
  2. Solve: 16 ( x + 1 ) = x 2 ( x + 1 )
  3. Solve: y 2 + 3 + 12 y 2 + 3 = 7    (Hint: Let y 2 + 3 = k and solve for k first and use the answer to solve y .)
  4. Solve for x : 2 x 4 - 5 x 2 - 12 = 0
  5. Solve for x :
    1. x ( x - 9 ) + 14 = 0
    2. x 2 - x = 3    (Show your answer correct to ONE decimal place.)
    3. x + 2 = 6 x    (correct to 2 decimal places)
    4. 1 x + 1 + 2 x x - 1 = 1
  6. Solve for x by completing the square: x 2 - p x - 4 = 0
  7. The equation a x 2 + b x + c = 0 has roots x = 2 3 and x = - 4 . Find one set of possible values for a , b and c .
  8. The two roots of the equation 4 x 2 + p x - 9 = 0 differ by 5. Calculate the value of p .
  9. An equation of the form x 2 + b x + c = 0 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 x and obtains the solutions 1 and -15. Determine the correct equation that was on theboard.
  10. Bjorn stumbled across the following formula to solve the quadratic equation a x 2 + b x + c = 0 in a foreign textbook.
    x = 2 c - b ± b 2 - 4 a c
    1. Use this formula to solve the equation:
      2 x 2 + x - 3 = 0
    2. Solve the equation again, using factorisation, to see if the formula works for this equation.
    3. Trying to derive this formula to prove that it always works, Bjorn got stuck along the way. His attempt his shown below:
      a x 2 + b x + c = 0 a + b x + c x 2 = 0 Divided by x 2 where x 0 c x 2 + b x + a = 0 Rearranged 1 x 2 + b c x + a c = 0 Divided by c where c 0 1 x 2 + b c x = - a c Subtracted a c from both sides 1 x 2 + b c x + ... Got stuck
      Complete his derivation.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Other chapter Q/A we can ask
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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