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Solution by the quadratic formula

It is not always possible to solve a quadratic equation by factorising and sometimes it is lengthy and tedious to solve a quadratic equation by completing the square. In these situations, you can use the quadratic formula that gives the solutions to any quadratic equation.

Consider the general form of the quadratic function:

f ( x ) = a x 2 + b x + c .

Factor out the a to get:

f ( x ) = a ( x 2 + b a x + c a ) .

Now we need to do some detective work to figure out how to turn [link] into a perfect square plus some extra terms. We know that for a perfect square:

( m + n ) 2 = m 2 + 2 m n + n 2

and

( m - n ) 2 = m 2 - 2 m n + n 2

The key is the middle term on the right hand side, which is 2 × the first term × the second term of the left hand side. In [link] , we know that the first term is x so 2 × the second term is b a . This means that the second term is b 2 a . So,

( x + b 2 a ) 2 = x 2 + 2 b 2 a x + ( b 2 a ) 2 .

In general if you add a quantity and subtract the same quantity, nothing has changed. This means if we add and subtract b 2 a 2 from the right hand side of [link] we will get:

f ( x ) = a ( x 2 + b a x + c a ) = a x 2 + b a x + b 2 a 2 - b 2 a 2 + c a = a x + b 2 a 2 - b 2 a 2 + c a = a x + b 2 a 2 + c - b 2 4 a

We set f ( x ) = 0 to find its roots, which yields:

a ( x + b 2 a ) 2 = b 2 4 a - c

Now dividing by a and taking the square root of both sides gives the expression

x + b 2 a = ± b 2 4 a 2 - c a

Finally, solving for x implies that

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

which can be further simplified to:

x = - b ± b 2 - 4 a c 2 a

These are the solutions to the quadratic equation. Notice that there are two solutions in general, but these may not always exists (depending on the sign ofthe expression b 2 - 4 a c under the square root). These solutions are also called the roots of the quadratic equation.

Find the roots of the function f ( x ) = 2 x 2 + 3 x - 7 .

  1. The expression cannot be factorised. Therefore, the general quadratic formula must be used.

  2. From the equation:

    a = 2
    b = 3
    c = - 7
  3. Always write down the formula first and then substitute the values of a , b and c .

    x = - b ± b 2 - 4 a c 2 a = - ( 3 ) ± ( 3 ) 2 - 4 ( 2 ) ( - 7 ) 2 ( 2 ) = - 3 ± 65 4 = - 3 ± 65 4
  4. The two roots of f ( x ) = 2 x 2 + 3 x - 7 are x = - 3 + 65 4 and - 3 - 65 4 .

Find the solutions to the quadratic equation x 2 - 5 x + 8 = 0 .

  1. The expression cannot be factorised. Therefore, the general quadratic formula must be used.

  2. From the equation:

    a = 1
    b = - 5
    c = 8
  3. x = - b ± b 2 - 4 a c 2 a = - ( - 5 ) ± ( - 5 ) 2 - 4 ( 1 ) ( 8 ) 2 ( 1 ) = 5 ± - 7 2
  4. Since the expression under the square root is negative these are not real solutions ( - 7 is not a real number). Therefore there are no real solutions to the quadratic equation x 2 - 5 x + 8 = 0 . This means that the graph of the quadratic function f ( x ) = x 2 - 5 x + 8 has no x -intercepts, but that the entire graph lies above the x -axis.

Khan academy video on quadratic equations - 2

Solution by the quadratic formula

Solve for t using the quadratic formula.

  1. 3 t 2 + t - 4 = 0
  2. t 2 - 5 t + 9 = 0
  3. 2 t 2 + 6 t + 5 = 0
  4. 4 t 2 + 2 t + 2 = 0
  5. - 3 t 2 + 5 t - 8 = 0
  6. - 5 t 2 + 3 t - 3 = 0
  7. t 2 - 4 t + 2 = 0
  8. 9 t 2 - 7 t - 9 = 0
  9. 2 t 2 + 3 t + 2 = 0
  10. t 2 + t + 1 = 0
  • In all the examples done so far, the solutions were left in surd form. Answers can also be given in decimal form, using the calculator. Read the instructions when answering questions in a test or exam whether to leave answers in surd form, or in decimal form to an appropriate number of decimal places.
  • Completing the square as a method to solve a quadratic equation is only done when specifically asked.

Mixed exercises

Solve the quadratic equations by either factorisation, completing the square or by using the quadratic formula:

  • Always try to factorise first, then use the formula if the trinomial cannot be factorised.
  • Do some of them by completing the square and then compare answers to those done using the other methods.
1. 24 y 2 + 61 y - 8 = 0 2. - 8 y 2 - 16 y + 42 = 0 3. - 9 y 2 + 24 y - 12 = 0
4. - 5 y 2 + 0 y + 5 = 0 5. - 3 y 2 + 15 y - 12 = 0 6. 49 y 2 + 0 y - 25 = 0
7. - 12 y 2 + 66 y - 72 = 0 8. - 40 y 2 + 58 y - 12 = 0 9. - 24 y 2 + 37 y + 72 = 0
10. 6 y 2 + 7 y - 24 = 0 11. 2 y 2 - 5 y - 3 = 0 12. - 18 y 2 - 55 y - 25 = 0
13. - 25 y 2 + 25 y - 4 = 0 14. - 32 y 2 + 24 y + 8 = 0 15. 9 y 2 - 13 y - 10 = 0
16. 35 y 2 - 8 y - 3 = 0 17. - 81 y 2 - 99 y - 18 = 0 18. 14 y 2 - 81 y + 81 = 0
19. - 4 y 2 - 41 y - 45 = 0 20. 16 y 2 + 20 y - 36 = 0 21. 42 y 2 + 104 y + 64 = 0
22. 9 y 2 - 76 y + 32 = 0 23. - 54 y 2 + 21 y + 3 = 0 24. 36 y 2 + 44 y + 8 = 0
25. 64 y 2 + 96 y + 36 = 0 26. 12 y 2 - 22 y - 14 = 0 27. 16 y 2 + 0 y - 81 = 0
28. 3 y 2 + 10 y - 48 = 0 29. - 4 y 2 + 8 y - 3 = 0 30. - 5 y 2 - 26 y + 63 = 0
31. x 2 - 70 = 11 32. 2 x 2 - 30 = 2 33. x 2 - 16 = 2 - x 2
34. 2 y 2 - 98 = 0 35. 5 y 2 - 10 = 115 36. 5 y 2 - 5 = 19 - y 2

Questions & Answers

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
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'?
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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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
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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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