# If Equations x^2+ax+b=0 & x^2+cx+d=0 have a common root and first equation have equal roots,solve Then prove that 2(b+d)=ac How to prove?solve

Apr 25, 2017

see below

#### Explanation:

note:

for this problem we will use the property of the sum and product of roots of a quadratic

that is

if $\text{ "alpha" " & " } \beta$ are the roots of

$p {x}^{2} + q x + r = 0$

then

$\alpha \beta = - \frac{q}{p}$

$\alpha \beta = \frac{r}{p}$

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${x}^{2} + a x + b = 0 - - - \left(1\right)$

${x}^{2} + c x + d = 0 - - - \left(2\right)$

let the common root be $\alpha$

for eqn$\left(1\right)$

$\alpha + \alpha = - a$

$\implies \alpha = - \frac{a}{2}$

$\text{ & } {\alpha}^{2} = b$

for the eqn$\left(2\right)$ let the second root be$\text{ } \beta$

then

$\alpha + \beta = - c$

$\alpha \beta = d$

$\implies \beta = \frac{d}{\alpha}$

$\therefore \alpha + \frac{d}{\alpha} = - c$

${\alpha}^{2} + d = \alpha \left(- c\right)$

$b + d = \left(- \frac{a}{2}\right) \left(- c\right)$

$\therefore 2 \left(b + d\right) = a c \text{ as reqd.}$