roots of equations

**abz** · July 10th 2008, 07:20 AM

Determine the roots of the equation
x2 – 3x – 8 = 0

**Moo** · July 10th 2008, 07:28 AM

Hello,

Originally Posted by abz

Determine the roots of the equation
x2 – 3x – 8 = 0

$x^2-3x-8=0$

Discriminant method :
$\Delta=9+32=41$

--> $x=\frac{3 \pm \sqrt{41}}{2}$

Completing the square method :
$x^2-3x-8=x^2-2 \cdot {\color{red}\frac 32} x-8=\left(x-\frac 32 \right)^2-\left(\frac 32\right)^2-8=\left(x-\frac 32 \right)^2-\frac 94-8$

$=\left(x-\frac 32 \right)^2-\frac{41}{4}=\left(x-\frac 32 \right)^2-\left(\frac{\sqrt{41}}{2}\right)^2$

Difference of 2 squares : $a^2-b^2=(a-b)(a+b)$

---> $x^2-3x-8=\left(x-\frac 32+\frac{\sqrt{41}}{2}\right)\left(x-\frac 32-\frac{\sqrt{41}}{2}\right)$

Therefore, $x-\frac 32+\frac{\sqrt{41}}{2}=0$ or $x-\frac 32-\frac{\sqrt{41}}{2}=0$

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