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Solving
equations
Clement
Radcliffe, Contributor
As we continue to review the solution
of quadratic equations using the formula
method, let us work the following
together.
Solve
the following
x²
- 9x + 14 = 0. Factorizing the Left
Hand Side
x²
- 9x + 14 = (x - 2)(x - 7)
ie. (x - 2)(x - 7) = 0
ie. x - 2 = 0, that is, x = 2 OR x
- 7 = 0, that is, x = 7
2x²
- x - 15 = 0
ie. (2x + 5)(x - 3) = 0
ie. 2x + 5 = 0, that is, x = -5/2
OR x - 3 = 0, that is, x = 6
2x
- x - 3 = 0
ie. (2x - 3)(x + 1) = 0
ie. x = 3/2 and -1
x²
+ x = 6
ie. x²
+ x - 6 = 0
ie. (x + 3)(x - 2) = 0
ie. x = - 3 and 2
Solve
y = 2x²
- 3x - 2 when y = 0
ie. y = 2x²
- 3x - 2 = 0
Factorizing
(2x + 1)(x-2) = 0
ie. x = -1/2 and 2
Most
quadratic equations cannot be solved
by factorization. Alternatively, the
FORMULA METHOD is used.
Please
be reminded that given the quadratic
equation ax²
+ bx + c = 0, where a, b
and c are constants, then it can be
shown that 
This
is the basis of the formula method
as x is found by substituting the
values of a, b and c into the formula.
Examples:
Express
2x²
= 3x + 1 in the form ax²
+ bx + c = 0 and find the values a
and c.
Given
that 2x²
= 3x + 1, then 2x²
- 3x - 1 = 0
By
comparing this equation with the required
form ax²
+ bx + c = 0
ie. a = 2, b = -3 and c = -1
Please be careful not to omit the
negative sign.
Answer: a = 2, b = -3 and c = -1
Solve 2x²
- 3x - 1 = 0. Using the Formula method:
From
the equation, a = 2, b = -3 and c
= -1
(Note
that the zero must be on the right
hand side).
Given
the formula:
Then
substituting
Let
us try another example.
Solve
the following equation using the quadratic
formula: 2x²
+ 2x - 8 = 3x - 6
2x²
+ 2x - 8 = 3x - 6
2x²
+ 2x - 3x - 8 + 6 = 0
2x²
- x - 2 = 0
Having expressed the equation into
the appropriate form, then a = 2,
b = -1 and c = -2
Unless
you are specifically directed, you
should attempt to use the factorization
method before the Formula method.
POINTS
TO NOTE
- Care
should always be taken in manipulating
the negative signs, as this provides
the greatest challenge in this method.
- The
± enables you to obtain two
roots.
- The
entire numerator is over 2a. A common
error is to use
separating -b. In other words, the
incorrect formula is 
is sometimes used.
- The
value within the square root should
always be positive. When this is
not so, it usually implies an error
in calculation. PLEASE CHECK YOUR
WORKING.
- If
the value within the square root
is negative, then the equation has
no real roots.
Please
find the solution of the quadratic
equations for homework.
1.
x²
+ 3x + 1 = 0
2.
2x²
- 6x - 1 = 0
3.
6x²
+ 11x = 10
4.
2x²
- 3x - 4 = 2 - 4x
* Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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