|
Quadratic
equations
Clement Radcliffe, Contributor
 | Jason,
a blind student of Meadowbrook High School, who is on the quiz team. - Ian Allen
Staff/Photographer |
You
were asked last week to attempt some practice examples. The following are the
solutions to some of these. Solve
the simultaneous equations: 2x
- 2y = 1 . . . (1)
7x
+ 2y = 17 . . . (2) Adding
equations (1) and (2) ie.
9x = 18 ie.
x = 2 Substitute x = 2 into equation (1) ie.
4 - 2y = 1
ie.
- 2y = 1 - 4 = -3
ie.
y = 3/2 Answer:
x = 2, y = 3/2 Solve
simultaneously: x
+ y = 7 . . . (1)
2x
+ y = 10 . . . (2) By
subtracting equation (2) from equation (1), and by following the method above,
I am sure that you can show that x = 3 and y = 4. x
- y = -5
3x
+ 2y = -5 Using
the elimination method, you need to convert the coefficients of one of the variables
to the same numerical value. In this case, it is most convenient to convert the
coefficients of y to +2 and -2. ie.
x - y = -5 . . . (1)
3x
+ 2y = -5 . . . (2) Multiply
equation (1) by 2. ie.
2x - 2y = -10 . . . (3) Adding
equations (2) and (3) ie.
5x = -15
ie. x = -15/3 = -3
ie. x = -3 Substituting into (1)
ie. 3 -
y = -5
ie.
- y = 3 -5 = -2
ie.
y = 2 Answer:
x = -3 , y = 2. Now
to a new topic. Solution
of quadratic equations The
following are the methods which are commonly used at this level. - Factorisation
- Formula
method
- Graphs
- Completing
the Square
We
will now begin with the factorisation method. Points
to note: - Quadratic
equations are expressed in the form ax²
+ bx + c = 0, where a, b and c are constants.
- The
factorisation method is used if, and only if, the expression ax²
+ bx + c can be factored.
- Given
the equation x²
+ 7x + 10 = 0, then by factoring the left hand side, you get
(x
+ 2 ) ( x + 5 ) = 0. If
(x + 2 )( x + 5 ) = 0 then
(x + 2) = 0, that is x = - 2 OR
x + 5 = 0, that is x = -5. Solutions
are x = -2 and x = -5. - Be
reminded that the solutions of the equation are the values which satisfy the equation.
These can be checked by substitution as follows:
If
x² + 7x
+ 10 = 0, then if x = -2, 4 -14 + 10 = 0. Similarly,
where x = -5, then 25 -35 + 10 = 0. The
equation is satisfied by both solutions. We
shall look at some examples. 1.
Solve 3x²
- 7x -6 = 0 Using
factorisation: 3x²
- 7x -6 = 0 ie.(3x
+ 2) (x - 3) =0
ie.3x
+ 2 = 0, that is, 3x = - 2
ie.x
= - 2/3 When
x - 3 = 0,
ie.x
= 3 Answer:
x = -2/3 and 3 2.
Solve the equation: 1
- 9x2 = 0 Factoring
using difference of two squares: 1
- 9x²
= (1 - 3x)(1 + 3x) ie.(1
- 3x)(1 + 3x) = 0 ie.1
- 3x = 0 or x = 1/3 1
+ 3x = 0 ...3x = -1 ie.x
= - 1/3. Answer:
x = 1/3 or - 1/3 Alternatively
1 -
9x² =
0
ie.9x²
= 1
x²
= 1/9
ie, x
= ± 1/3. 3.
Solve the equation: 3(x +2)²
= 7(x + 2) (June 1990, No. 3a) 3(x
+2)2 = 7(x + 2) Clearing the brackets: ie.
3(x² +
4x + 4) = 7x + 14. ie.
3x² +
12x + 12 = 7x + 14. ie.3x²
+ 12x -7x + 12 - 14 = 0. ie.3x²
+ 5x -7x - 2 = 0. Factorizing: ie.(3x
- 1)(x + 2) = 0 ie.3x
-1 = 0, that is, x = 1/3 OR
x + 2 = 0, that is, x = -2. Answers
are x = 1/3 and -2. Now
that you are comfortable with solving simultaneous linear equations and some quadratic
equations, you can now attempt the following for homework. x²
- 9x + 14 = 0 2x²
- x - 15 = 0 2x²
- x - 3 = 0 x²
+ x = 6 Solve
the equation: y = 2x²
- 3x - 2 when y = 0 It
is important that you practise questions from CXC past papers. Since
this is the final lesson for this term, let me use the opportunity to wish you
all a happy holiday. May you use the time to be fully charged for the challenges
ahead. Clement
Radcliffe teaches at Glenmuir High School. | 
