The quadratic equation $3 x^{2} - 9 x + b = 0$ $3x^2 -9x +b =0$ has roots $α$ $alpha$ and $α + 2$ $alpha+2$ , find $b$ $b$ ?

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Answer 1 · 2017-04-01T18:24:31

$b = \frac{15}{4}$ $b=15/4$

Suppose the roots of the general quadratic equation:

$a x^{2} + b x + c = 0$ $ax^2+bx+c = 0$

are $α$ $alpha$ and $β$ $beta$ , then using the root properties we have:

$"sum of roots" \ \ \ \ \ \= alpha+beta = -b/a$
$"product of roots" = alpha beta \ \ \ \ = c /a$

So for the given quadratic with roots $alpha$ and $beta$ :

$3x^2 -9x +b =0$

we know that:

$alpha+beta = -(-9)/3=3 \ \ \$ ; and $\ \ \ alpha beta = b/3$

But we also know that $beta=alpha + 2$

Hence,

$alpha+beta= = alpha+(alpha+2) = 2alpha+2 => 2alpha+2=3$
$:. alpha = 1/2$

and:

$alpha beta= = alpha(alpha+2) = 1/2(1/2+2) = 5/4$
$:. 5/4 = b/3 => b=15/4$

The quadratic equation 3x^2 -9x +b =0 3x2−9x+b=0 3x^2 -9x +b =0 has roots alphaαalpha and alpha+2α+2alpha+2, find bbb?