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

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ax^2 + 8x + 9 =0

==> a=a    b= 8     and c = 9

if the quadratic equation has one root, then we know that delta should be zero:

Delta = b^2 - 4ac

==> 0 = 8^2 - 4*a*9

==> 64 - 36a = 0

Subtract 64 from both...

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ax^2 + 8x + 9 =0

==> a=a    b= 8     and c = 9

if the quadratic equation has one root, then we know that delta should be zero:

Delta = b^2 - 4ac

==> 0 = 8^2 - 4*a*9

==> 64 - 36a = 0

Subtract 64 from both sides:

==> -36a = -64

Divide by -36

==> a = -64/-36

==> a = 16/9

Then the values of a in order to have one root is a= 16/9

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The roots of a quadratic equation are given by the expressions [–b + sqrt (b^2 – 4ac)]/2a and [–b - sqrt (b^2 – 4ac)]/2a. Now in the question we have to find the value of a for which the quadratic equation has two equal roots. So we need to have:

[–b + sqrt (b^2 – 4ac)]/2a = [–b - sqrt (b^2 – 4ac)]/2a

=> sqrt (b^2 – 4ac) = 0

=> (b^2 – 4ac) =0

=> b^2 = 4ac

=> a = b^2 / 4c

as b = 8 and c = 9

=> a = 8^2 / 4*9

=> a = 16/9

Therefore to satisfy the given condition a should be equal to 16/9

Approved by eNotes Editorial Team