what is the conditioncondition for equations 3x^2 - 2x- 1 = 0 and ( m+ 2 )x^2 - 3x + m = 0 to have one equal root .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we'll have to calculate the roots of the 1st determined equation:3x^2 - 2x- 1 = 0

We'll use quadratic formula:

x1 = [2+sqrt(4 + 12)]/6

x1 = (2+4)/6

x1 = 1

x2 = (2-4)/6

x2 = -1/3

Now, we'll impose that x = 1 to be the root of the equation ( m + 2 )x^2 - 3x + m = 0

We'll substitute x by 1 in the given equation:

(m+2)*1^2 - 3*1 + m = 0

We'll remove the brackets:

m + 2 - 3 + m = 0

2m - 1 = 0

m = 1/2

Now, we'll impose that x = -1/3 to be the root of the equation ( m + 2 )x^2 - 3x + m = 0

(m+2)/9 + 1 + m = 0

We'll multiply by 9:

m + 2 + 9 + 9m = 0

10m + 11 = 0

m = -11/10

The condition for equations 3x^2 - 2x- 1 = 0 and ( m+ 2 )x^2 - 3x + m = 0 to have one equal root  is that m has to have the following values: {-11/10 ; 1/2}.

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