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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

explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
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
please help me prove quadratic formula
Darius
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
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
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
Other chapter Q/A we can ask
Moahammedashifali Reply

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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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