Algeb

**srirahulan** · May 21st 2012, 07:50 PM

If $x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$

**Prove It** · May 21st 2012, 08:27 PM

Originally Posted by srirahulan

If $x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$

The roots of the first equation are

$\displaystyle \begin{align*} x &= \frac{-ab \pm \sqrt{(ab)^2 - 4(1)(c)}}{2(1)} \\ &= \frac{-ab \pm \sqrt{a^2b^2 - 4c}}{2} \end{align*}$

and the roots of the second equation are

$\displaystyle \begin{align*} x &= \frac{-ac \pm \sqrt{(ac)^2 - 4(1)(b)}}{2(1)} \\ &= \frac{-ac \pm \sqrt{a^2c^2 - 4b}}{2} \end{align*}$

Since they share a root, there is some value such that

$\displaystyle \begin{align*} \frac{-ab \pm \sqrt{a^2b^2 - 4c}}{2} &= \frac{-ac \pm \sqrt{a^2c^2 - 4b}}{2} \\ -ab \pm \sqrt{a^2b^2 - 4c} &= -ac \pm \sqrt{a^2c^2 - 4b} \end{align*}$

If $\displaystyle \begin{align*} b = c \end{align*}$ , we have

$\displaystyle \begin{align*} LHS &= -ab \pm \sqrt{a^2b^2 - 4c} \\ &= -ab \pm \sqrt{a^2b^2 - 4b} \\ \\ RHS &= -ac \pm \sqrt{a^2c^2 - 4b} \\ &= -ab \pm \sqrt{a^2b^2 - 4b} \\ &= LHS \end{align*}$

So that means for the two equations to share a root, it's possible that $\displaystyle \begin{align*} b = c \end{align*}$ .

**srirahulan** · May 22nd 2012, 12:25 AM

I can't understand you statements.pls reply,

**Wilmer** · May 22nd 2012, 04:45 AM

Originally Posted by srirahulan

....are the roots of $a(b+c)x^2+(b+c)-abc=0$

k = [abc - (b+c)] / [a(b+c)]
x = +- SQRT(k)

**Wilmer** · May 22nd 2012, 05:43 AM

Originally Posted by srirahulan

If $x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0$ quadratic equations have a common root.prove b=c or the other roots are the roots of $a(b+c)x^2+(b+c)-abc=0$

x^2 + abx + c = 0 [1]
x^2 + acx + b = 0 [2]

Roots of [1] = u,v
Roots of [2] = u,w
Then:
ab = u + v
c = uv : v = c/u
ab = u + c/u : a = (u + c/u) / b [3]

Similarly: a = (u + b/u) / c [4]

[3][4]: (u + c/u) / b = (u + b/u) / c
Simplify:
bu^2 + b^2 = cu^2 + c^2
b = c

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