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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
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Kyle
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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