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Solving
quadratic equations
Clement
Radcliffe, Contributor
Let
me hope that you had a merry Christmas
and are poised for a prosperous New
Year. This I expect will be with respect
to your efforts in mathematics.
I
will now proceed to present some reminders
of how to solve quadratic equations.
*
Solve 2x²
- 6x - 1 = 0
From
the equation, a = 2, b = - 6 and c
= - 1.
Answer:
x =3.16 and - 0.16
*
Solve 2x²
- 3x - 4 = 2 - 4x
First
simplify the equation:
2x²
- 3x + 4x - 4 - 2 = 0
2x²
+ x - 6 = 0 Factorising
(2x
- 3)(x + 2) = 0
2x
- 3 = 0, that is x = 3
2
And
x + 2 = 0, that is x = -2.
Answer:
x = 3/2 and -2
Quadratic
equations may also be solved using
the method of completion of squares.
COMPLETION
OF SQUARES
Given
the equation x²
+ bx + c = 0, the aim is to convert
the equation to the form (x + d)²
= k, where d and k are constants.
You are therefore required to CONVERT
the left hand side to a PERECT SQUARE
of form (x + d)².
Given the form (x + d)²
= k, then x is found by determining
the square root of both sides.
(x
+ d)²
= k
(x
+ d) = ± k
x
= - d ± k
As
was the case previously, the ±
(plus or minus) will enable you to
find the two values of x.
Example:
Solve:
x²
+ 6x - 5 = 0, using completion of
squares.
Given
x²
+ 6x - 5 = 0, the first critical step
is to transfer the constant to the
right hand side (RHS):
x²
+ 6x = 5
This
is followed by making the left hand
side a perfect square. This is based
on the following equation:
(x
+ a) ²
= x²
+ 2ax + a²
.
Given
x²
+ 2ax, then a²,
the square of half the coefficient
of x, must be added to complete the
square.
Given
x²
+ 6x, then (6/2)²
or 3²,
the square of half the coefficient
of x is required to make
the left hand side a perfect square.
x²
+ 6x + 9 = 5 + 9 = 14
3²
or 9 is added to both sides of the
equation so that the value of x remains
unchanged.
x²
+ 6x + 9 = 14. (Factorising the LHS)
(x
+ 3)²
= 14. Find the square root of both
sides
x
+ 3 = ± 14 = ± 3.74
x
+ 3 = 3.74, that is x = 0.74
OR
x + 3 = -3.74, that is x = - 6.74
Answer
x = 0.74 or - 6.74
I
am sure you won't mind us practising
the following together.
Examples:
(1)
Express x²
- 8x = 1 in the form (x + a)²
= b.
As
( 8/2 )²
= 16, then 16 must be added to both
sides to make the LHS a perfect square.
x²
- 8x + 16 = 1 + 16 Factorising the
LHS
(x
- 4)²
= 17.
You
may expand to check your answers.
(2)
Find the values to be added to the
L.H.S. to make it a perfect square.
(a)
x²
- 12x = 1
(b)
y²
+ 9x = 4
Your
answers should be 36 and 81 respectively.
If this is not so, then please review
the method.
4
Now
let us solve the above examples together.
(a)
Solve x²
- 8x = 1
If
x²
- 8x = 1
Completing
the squares:
x²
- 8x + 16 = 1 + 16
(x
- 4)²
= 17
x
- 4 = ± 17
x
= 4 ± 4.12
x
= 8.12 and -.12
(b)
Solve x²
- 12x = 1
Since
x²
- 12x = 1
Completing
the squares:
x²
- 12x + 36 = 1 + 36
(x
- 6)²
= 37
x
- 6 = ± 37
x
= 6 ± 6.08
x
= 12.08 and -.08
Set
out below is your homework.
Solve
the following equations, using the
completion of squares method.
(1)
x²
+ 9x - 4 = 0
(2)
x²
+ 8x - 9 = 0
(3)
x²
= 7x + 5
*
Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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