Quadratic .If a quadratic equation ax^2 + 8x + 9 = 0 has two equal roots what is the value of a?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the information that the quadratic equation has two equal roots, hence its determinant, `Delta = b^2 - 4ac ` needs to be equal to zero, such that:

`Delta = 0 => b^2 - 4ac = 0`

You need to identify the coefficients of equation, such that:

 

`a = a, b = 8, c = 9 `

`Delta = 8^2 - 4*a*9 => 64 - 36a = 0=> -36a = -64 => a = 64/36`

`=> a = 16/9`

Hence, evaluating the leading coefficient, under the given conditions, yields `a = 16/9.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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A quadratic equation has always 2 roots. When you say that the equation has "only one root", you are wrong. In fact, the equation has 2 equal roots.

ax^2 + 8x + 9 = 0

We'll use Viete's relations between coefficients and roots:

x1 + x2 = -8/a (1)

x1*x2 = 9/a (2)

But x1 = x2, because the roots are equal

x1 + x1 = -8/a

2x1 = -8/a

x1 = -4/a

x1^2 = 9/a => (-4/a)^2 = 9/a

We'll square raise and we'll subtract 9/a both sides:

16/a^2 - 9/a = 0

We'll eliminate the denominator:

16 - 9a = 0

9a = 16

We'll divide by 9 both sides:

a = 16/9

So, for the quadratic equatino to have 2 equal roots, the value of the coefficient a is 16/9.

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