QUADRATIC
EXPRESSIONS &
EQUATIONS
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LEARNING OUTCOMES:
At the end of the lesson, the students should be
able to:
(d)Define quadratic equation
(e)Solve quadratic equations by
(i) Factorization
(ii) Completing the square
(iii) Quadratic formula.
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Consider a rectangular field with sides of (x-4)
meter by x meter.
The area of the field is
A = ( x − 4)( x)
= x2 − 4x
The expression is known as
QUADRATIC EQUATION
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Expression and Quadratic Equation
• Quadratic Equation
- General form of quadratic equation:
ax2 + bx + c = 0
Highest degree – 2
a, b, c – constant and a ≠ 0
x∈R
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The methods of solving the quadratic
equation are:
• Factorizing (this method can only be used
for quadratic equation that can be factorized)
• Completing the square
• Using the quadratic formula
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(1) Factorization Method
Example:
Solve the following equation using
factorization.
(i) x2 – 7x + 12 = 0
Solution
(x - 3) (x - 4) = 0
x – 3 = 0 or x – 4 = 0
x = 3 or x = 4
Solution set: { x: x = 3 or x = 4}
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(ii ) 2 x + 5 x + 3 = 0
2
solution
(2 x + 3)( x + 1) = 0
2 x + 3 = 0 or x + 1 = 0
3
x = − or x = −1
2
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(iii ) x - 25 = 02
solution
x = 25 2
x = ±5
x = −5 or x = 5
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(2) Completing the square
STEPS:
1) Coefficient of x2 = 1
2
2) Add both sides with b
2
3) The left side is factorized into complete
square
4) Solve by taking the square root both sides
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Example
Solve the following equations by completing the
square.
1) x2 + 6x - 3 = 0
2) 2x2 – 4x – 1 = 0
3) -4x2 + 16x – 4 = 0
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(3) Using the quadratic formula
b b 4ac 2
x
2a
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Example
Solve the following equations by using the
quadratic formula.
1) 4x2 + 9x – 2 = 0
2) x2 – x – 1 = 0
3) 9s2 + 2 = 12s
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EXERCISE
1. Solve the following quadratic equations by
factorization
5 w2 2
x 6 0 w
a) x b) 3
2. Solve the following quadratic equations by
completing the square
a) 2 x 3x 1 0
2
b) 3x 2 1 5 x
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3. Solve the following quadratic equations
by quadratic formula
a) x 2 x 5 0
2
b) 2 x 4 3x 2
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Nature of the Roots
of a Quadratic
Equation
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LEARNING OUTCOMES:
At the end of the lesson, students will be
able to
(d) Recognize the type of roots of a
quadratic equation
(b) Relate the roots and with
coefficients of quadratic equation
(c) Form quadratic equations using
identities
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The nature of the roots of the ax2 + bx + c = 0 is
determined by the expression
b2 – 4ac which is known as the discriminant
There are three types of root for quadratic equation
1) If b2 – 4ac > 0 two distinct real roots
2) If b2 – 4ac = 0 two equal real roots
3) If b2 – 4ac < 0 two complex roots
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Example
1. Determine the nature of the roots of the
following quadratic equations
a) x2 - 8x + 16 = 0
b) x2 – 4x + 6 = 0
c)3x2 – 9x + 5 = 0
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2.Find the value of p if 2x2 – 8x + p=0 has equal
roots.
3.Show that the quadratic equation
kx2 + 5x + 5 = k has two real roots for all values
of k
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The relationship between
the roots with the
coefficients of quadratic
equation
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• The quadratic equation with roots α and β
can be written in the form
(x - α) ( x - β ) = 0
that is x2 - (α + β) x + αβ = 0> .....(1)
• the equation ax2 + bx + c = 0 , a ≠ 0, can be
written as
b c
x
2
x 0……….. (2)
a a
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• Compare (1) = (2), hence
b
-(α + β) =
a
b
(α + β) = - (known as sum of roots)
a
c
αβ = (known as product of roots)
a
Or the equation can also be written as :
X 2 –(sum of roots)x +(product of roots)= 0
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The important identities:
2
2 2 2
2 2
3
3 3
2
3 3 2
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Example
1. The roots of the equation
2x2 + 5x – 4 = 0 are α and β . Find the values
of
a) α2 + β2
b) (α - β)2
c) α3 - β3
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2. The roots of the equation 3x2 – 2x – 1 = 0
are α and β. Find the equation with the
1
following roots ( )and ( 1 )
3. If α and β are roots of the equation
x2 - 3x +1 = 0, find
a) the value of (2α + β)(α +2β)
b) a quadratic equation whose roots are
2 2
and
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3. Given that one root of the equation
4x2 + 3x = k is two times the other, find the
value of k
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EXERCISE
1. Find the values of p so that
px2+px+3x+p = 0 has two equal real
roots
2. Prove that the equation
(k-4)x2+4x+k = 0 has two real roots for
all values of k
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3. If α and β are the roots of the equation
3x2 – 2x – 5 = 0. Find the equation
whose new roots are and
4. Given that α and β are the roots of the
quadratic equation 2x2 + 5x – 4 = 0. Write
down the values of
a) α + β
2 2
b)
2
2
1 1
c) α + β
3 3
d)
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