If α, β, are the roots of the quadratic equation, then the form of the quadratic equation as
x^{2} – (α + β)x + αβ = 0
Where,
α + β = sum of roots
αβ = product of roots
Problem 1 :
Form the equation whose roots are 1 and -5.
Solution :
Let the given roots be α and β
Sum of the roots = α + β
Product of roots = αβ
α = 1, β = -5
α + β = 1 + (-5) = 1 – 5 = -4
αβ = 1 × (-5) = -5
x^{2} – (α + β)x + αβ = 0
x^{2} – (-4)x + (-5) = 0
x^{2} + 4x - 5 = 0
So, the equation is x^{2} + 4x - 5 = 0.
Problem 2 :
Form the equation whose roots are
(√3 - 2)/2 and (√3 + 2)/2
Solution :
Roots are (√3 - 2)/2 and (√3 + 2)/2
Sum of the roots = α + β
Product of roots = αβ
α = (√3 - 2)/2, β = (√3 + 2)/2
x^{2} - √3 x + (-1/4) = 0
4x^{2} - √3 x -1 = 0
Problem 3 :
If 1 – i and 1 + i are the roots of the equation
x^{2} + ax + b = 0
where a, b ∈ r, then find the values of a and b.
Solution :
α = 1 - i and β = 1 + i
Creating quadratic equation,
α + β = 1 - i + 1 + i ==> 2
α β = (1 - i) (1 + i) ==> 1 - i^{2} ==> 1 - (-1) ==> 2
x^{2} - 2x + 2 = 0
Comparing the given equation with
x^{2} + ax + b = 0
a = -2 and b = 2
Problem 4 :
The roots of a quadratic equation are -2 and -6. The minimum point of the graph of its related function is at (-4, -2). Sketch the graph of the function.
Solution :
α = -2 and β = -6
α + β = -2 - 6 ==> -8
α β = -2 (-6) ==> 12
The quadratic function will be,
y = a(x^{2} - (-8)x + 12)
y = a(x^{2} + 8x + 12)
Since the quadratic function is having a minimum point (-4, -2), the will pass through this point.
-2 = a((-4)^{2} + 8(-4) + 12)
-2 = a(16 - 32 + 12)
-2 = a(-4)
a = 1/2
By applying the value of a, we get
y = (1/2)(x^{2} + 8x + 12)
So, the required quadratic function is
y = (1/2)(x^{2} + 8x + 12)
Problem 5 :
The roots of a quadratic equation are -6 and 0. The minimum point of the graph of its related function is at (-3, 4). Sketch the graph of the function.
Solution :
α = -6 and β = 0
α + β = -6 + 0 ==> -6
α β = -6 (0) ==> 0
The quadratic function will be,
y = a(x^{2} - (-6)x + 0)
y = a(x^{2} + 6x)
Since the quadratic function is having a minimum point (-3, 4), the will pass through this point.
4 = a((-3)^{2} + 6(-3))
4 = a(9 - 18)
4 = a(-9)
a = -4/9
By applying the value of a, we get
y = (-4/9)(x^{2} + 6x)
So, the required quadratic function is
y = (-4/9)(x^{2} + 6x)
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM