Any quadratic equation will be in the form
y = ax^{2} + bx + c
It has two roots or zeroes α and β.
Using the above roots, quadratic equation can be created using the formula given below.
x^{2} - (α + β) x + α β = 0
Here, α + β = Sum of roots
α β = Product of roots
From each problem, find a quadratic equation
with those numbers as its solutions.
Problem 1 :
2, 5
Solution :
Given, 2, 5
α = 2, β = 5
x^{2} – (α + β)x + αβ = 0
x^{2} – (2 + 5)x + (2 × 5) = 0
x^{2} – 7x + 10 = 0
Problem 2 :
3, 6
Solution :
Given, 3, 6
α = 3, β = 6
x^{2} – (α + β)x + αβ = 0
x^{2} – (3 + 6)x + (3 × 6) = 0
x^{2} – 9x + 18 = 0
Problem 3 :
20, 2
Solution :
Given, 20, 2
α = 20, β = 2
x^{2} – (α + β)x + αβ = 0
x^{2} – (20 + 2)x + (20 × 2) = 0
x^{2} – 22x + 40 = 0
Problem 4 :
13, 1
Solution :
Given, 13, 1
α = 13, β = 1
x^{2} – (α + β)x + αβ = 0
x^{2} – (13 + 1)x + (13 × 1) = 0
x^{2} – 14x + 13 = 0
Problem 5 :
4, 4
Solution :
Given, 4, 4
α = 4, β = 4
x^{2} – (α + β)x + αβ = 0
x^{2} – (4 + 4)x + (4 × 4) = 0
x^{2} – 8x + 16 = 0
Problem 6 :
0, 9
Solution :
Given, 0, 9
α = 0, β = 9
x^{2} – (α + β)x + αβ = 0
x^{2} – (0 + 9)x + (0 × 9) = 0
x^{2} – 9x + 0 = 0
Problem 7 :
-2, -5
Solution :
Given, -2, -5
α = -2, β = -5
x^{2} – (α + β)x + αβ = 0
x^{2} – (-2 + (-5))x + (-2 × -5) = 0
x^{2} + 7x + 10 = 0
Problem 8 :
-4, 11
Solution :
Given, -4, 11
α = -4, β = 11
x^{2} – (α + β)x + αβ = 0
x^{2} – ((-4) + 11)x + (-4 × 11) = 0
x^{2} – 7x - 44 = 0
Problem 9 :
3, -1
Solution :
Given, 3, -1
α = 3, β = -1
x^{2} – (α + β)x + αβ = 0
x^{2} – (3 + (-1))x + (3 × -1) = 0
x^{2} – 2x - 3 = 0
Problem 10 :
3/4, 1/4
Solution :
Given, 3/4, 1/4
α = 3/4, β = 1/4
x^{2} – (α + β)x + αβ = 0
x^{2} – (3/4 + 1/4)x + (3/4 × 1/4) = 0
x^{2} – (4/4)x + 3/16 = 0
Multiply 16 on each sides.
16x^{2} – 16x + 3 = 0
Problem 11 :
5/8, 5/7
Solution :
α = 5/8
β = 5/7
α + β = 5/8 + 5/7
= 5/8 × 7/7 + 5/7 × 8/8
= 35/56 + 40/56
= 75/56
α × β = 5/8 × 5/7
= 25/56
x^{2} – (α + β)x + αβ = 0
x^{2} – (75/56)x + 25/56 = 0
Multiply 56 on each sides.
56x^{2} – 75x + 25 = 0
Problem 12 :
1/2, 1/3
Solution :
α = 1/2
β = 1/3
α + β = 1/2 + 1/3
= 1/2 × 3/3 + 1/3 × 2/2
= 3/6 + 2/6
= 5/6
α × β = 1/2 × 1/3
= 1/6
x^{2} – (α + β)x + αβ = 0
x^{2} – (5/6)x + 1/6 = 0
Multiply 6 on each sides.
6x^{2} – 5x + 1 = 0
Problem 13 :
1/2, 2/3
Solution :
α = 1/2
β = 2/3
α + β = 1/2 + 2/3
= 1/2 × 3/3 + 2/3 × 2/2
= 3/6 + 4/6
= 7/6
α × β = 1/2 × 2/3
= 2/6
x^{2} – (α + β)x + αβ = 0
x^{2} – (7/6)x + 2/6 = 0
Multiply 6 on each sides.
6x^{2} – 7x + 2 = 0
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM