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Introduction

In grade 10, the basics of solving linear equations, quadratic equations, exponential equations and linear inequalities were studied. This chapter extends on that work. We look at different methods of solving quadratic equations.

Solution by factorisation

How to solve quadratic equations by factorisation was discussed in Grade 10. Here is an example to remind you of what is involved.

Solve the equation 2 x 2 - 5 x - 12 = 0 .

  1. This equation has no common factors.

  2. The equation is in the required form, with a = 2 , b = - 5 and c = - 12 .

  3. 2 x 2 - 5 x - 12 has factors of the form:

    ( 2 x + s ) ( x + v )

    with s and v constants to be determined. This multiplies out to

    2 x 2 + ( s + 2 v ) x + s v

    We see that s v = - 12 and s + 2 v = - 5 . This is a set of simultaneous equations in s and v , but it is easy to solve numerically. All the options for s and v are considered below.

    s v s + 2 v
    2 -6 -10
    -2 6 10
    3 -4 -5
    -3 4 5
    4 -3 -2
    -4 3 2
    6 -2 2
    -6 2 -2

    We see that the combination s = 3 and v = - 4 gives s + 2 v = - 5 .

  4. ( 2 x + 3 ) ( x - 4 ) = 0
  5. If two brackets are multiplied together and give 0, then one of the brackets must be 0, therefore

    2 x + 3 = 0

    or

    x - 4 = 0

    Therefore, x = - 3 2 or x = 4

  6. The solutions to 2 x 2 - 5 x - 12 = 0 are x = - 3 2 or x = 4 .

It is important to remember that a quadratic equation has to be in the form a x 2 + b x + c = 0 before one can solve it using the following methods.

Solve for a : a ( a - 3 ) = 10

  1. Remove the brackets and move all terms to one side.

    a 2 - 3 a - 10 = 0
  2. ( a + 2 ) ( a - 5 ) = 0
  3. a + 2 = 0

    or

    a - 5 = 0

    Solve the two linear equations and check the solutions in the original equation.

  4. Therefore, a = - 2 or a = 5

Solve for b : 3 b b + 2 + 1 = 4 b + 1

  1. 3 b ( b + 1 ) + ( b + 2 ) ( b + 1 ) ( b + 2 ) ( b + 1 ) = 4 ( b + 2 ) ( b + 2 ) ( b + 1 )
  2. The denominators are the same, therefore the numerators must be the same.

    However, b - 2 and b - 1

  3. 3 b 2 + 3 b + b 2 + 3 b + 2 = 4 b + 8 4 b 2 + 2 b - 6 = 0 2 b 2 + b - 3 = 0
  4. ( 2 b + 3 ) ( b - 1 ) = 0 2 b + 3 = 0 o r b - 1 = 0 b = - 3 2 o r b = 1
  5. Both solutions are valid

    Therefore, b = - 3 2 or b = 1

Solution by factorisation

Solve the following quadratic equations by factorisation. Some answers may be left in surd form.

  1. 2 y 2 - 61 = 101
  2. 2 y 2 - 10 = 0
  3. y 2 - 4 = 10
  4. 2 y 2 - 8 = 28
  5. 7 y 2 = 28
  6. y 2 + 28 = 100
  7. 7 y 2 + 14 y = 0
  8. 12 y 2 + 24 y + 12 = 0
  9. 16 y 2 - 400 = 0
  10. y 2 - 5 y + 6 = 0
  11. y 2 + 5 y - 36 = 0
  12. y 2 + 2 y = 8
  13. - y 2 - 11 y - 24 = 0
  14. 13 y - 42 = y 2
  15. y 2 + 9 y + 14 = 0
  16. y 2 - 5 k y + 4 k 2 = 0
  17. y ( 2 y + 1 ) = 15
  18. 5 y y - 2 + 3 y + 2 = - 6 y 2 - 2 y
  19. y - 2 y + 1 = 2 y + 1 y - 7

Questions & Answers

find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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