- #1

mcsun

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## Homework Statement

Solve the equation ax

^{2}+ bx + c = 0. The following are the given information:

(a) a = 1

(b) One root is twice the other root.

(c) c = 2

You are NOT required to solve the problem but to decide whether:

1) The given information is all needed to solve the problem. In this case write "A" as your solution.

2) The total amount of information is insufficient to solve the problem. If so, write "I" as your solution.

3) The problem can be solved without using one or more of the given pieces of information. In this case, write the letter(s) corresponding to the items not needed.

Solution provided by the textbook:

Answer (I)

## Homework Equations

General Quadratic Equation:

ax

^{2}+ bx + c = 0

Let [tex]\alpha[/tex] and [tex]\beta[/tex] be the roots of the equation ax

^{2}+ bx + c = 0

Sum of Roots = ([tex]\alpha[/tex] + [tex]\beta[/tex]) = - [[tex]\frac{b}{a}[/tex]]

Product of Roots = ([tex]\alpha[/tex][tex]\beta[/tex]) = [tex]\frac{c}{a}[/tex]

## The Attempt at a Solution

My answer is (A).

Why? I shall poof:

The equation: x

^{2}+ bx + 2 = 0

Firstly determine the value of "b":

x

^{2}+ bx + 2 = 0

*[ax*

^{2}+ bx + c = 0]Sum of its Roots = ([tex]\alpha[/tex] + [tex]\beta[/tex]) = - [[tex]\frac{b}{a}[/tex]] = - [[tex]\frac{b}{1}[/tex]] = - b

Product of its Roots = ([tex]\alpha[/tex][tex]\beta[/tex]) = [tex]\frac{c}{a}[/tex] = [tex]\frac{2}{1}[/tex] = 2

Since the question states that "One root is twice the other", therefore...

[tex]\beta[/tex] = 2[tex]\alpha[/tex]

Hence,

Sum of its Roots = ([tex]\alpha[/tex] + 2[tex]\alpha[/tex]) = 3([tex]\alpha[/tex]) = -b

Product of its Roots = ([tex]\alpha[/tex])(2[tex]\alpha[/tex]) = 2([tex]\alpha[/tex])

^{2}= 2

Since,

2([tex]\alpha[/tex])

^{2}= 2

[tex]\alpha[/tex] = 1 ---> (1)

3([tex]\alpha[/tex]) = -b ---> (2)

Substitute (1) into (2):

b = -3

Now substitute [b = -3] into the original quadratic equation:

x

^{2}- 3x + 2 = 0

(x - 2)(x - 1) = 0

Hence, x = 2 of x = 1

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Question:

Is the above answer & proof of mine correct? Or am I absolutely wrong?

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**TAKE NOTE:**

3([tex]\alpha[/tex]) is NOT EQUAL to 3^([tex]\alpha[/tex])

3([tex]\alpha[/tex]) is NOT EQUAL to 3^([tex]\alpha[/tex])

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