Solved: "If both roots of the equation ax2−2bx+5=0 are..."

High School
Algebra
Algebra I
Quadratic function and equation

chingli20013a1

Answered question

2022-04-23

If both roots of the equation

a x^{2} - 2 b x + 5 = 0

are

α

and roots of the equation

x^{2} - 2 b x - 10 = 0

are

α

and

β

Answer & Explanation

dagars5nx

Beginner2022-04-24Added 12 answers

Step 1
From the first equation,

α^{2} = \frac{5}{a}, 2 α = \frac{2 b}{a}

and we obtain

α = \frac{5}{b}

From the second equation

α + β = 2 b, α β = - 10

Then

β = - 2 b

and

\frac{5}{b} = 4 b

i.e.

b^{2} = \frac{5}{4} .

Then

α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = 4 b^{2} + 20 = 25 .

Dexter Conner

Beginner2022-04-25Added 15 answers

Step 1
$b^{2} = 5 a$ then $α = \frac{b}{a}$ Next we have $α + β = 2 b$ and $α β = - 10$ Then
$α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = 4 b^{2} + 20 = 20 a + 20$
Next we have
$β = \frac{- 10}{α} = 2 b - α \Rightarrow - \frac{10 a}{b} = 2 b - \frac{b}{a}$
$\Rightarrow - 10 a^{2} = 2 a b^{2} - b^{2} \Rightarrow - 20 a^{2} = - 5 a \Rightarrow a = \frac{1}{4},$ as $a \neq 0$
Finally from (1) we get $α^{2} + β^{2} = 25$

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