Answer:
c = 27/5
Step-by-step explanation:
Given data:
Standard form of the quadratic equation
a x² + b x + c = 5 x² - 12 x + c = 0
and relation between the roots x₂ = 3 x₁
From which we conclude the following:
a = 5 , b = -12 and c = ?
To solve this problem we will use the Vieta's formulas, they read:
x₁ + x₂ = - b/a and x₁ · x₂ = c/a
First we will replace x₂ = 3 x₁ in the first formula and get:
x₁ + 3 x₁ = - (-12)/5 => 4 x₁ = 12/5 => x₁ = (12/5) : 4 = 3/5
x₁ = 3/5
Now we will replace x₁ = 3/5 in the relation x₂ = 3 x₁ and get:
x₂ = 3 · 3/5 = 9/5
x₂ = 9/5
Now we will replace value for x₁ = 3/5 and x₂ = 9/5 in the second formula
x₁ · x₂ = c/a and get:
(3/5) (9/5) = c/5 => 27/25 = c/5 => c = (27/25) : 5 = 27/5
c = 27/5
Now the quadratic equation reads:
5 x² - 12 x + 27/5 = 0 / · 5
We will multiply whole equation with number 5 and get:
25 x² - 60 x + 27 = 0
Prove:
Every quadratic equation can be factorized as follows:
a (x - x₁) (x - x₂) = a x² + b x + c
we calculated x₁ = 3/5 and x₂ = 9/5
5 (x - (3/5)) (x - (9/5)) = (5 x - 3) (x - (9/5)) = 5 x² - 9 x - 3 x + 27/5 =
= 5 x² - 12 x + 27/5
God with you!!!
