Factor

**habibixox** · September 15th 2011, 02:17 PM

3x^2+5x-7

I want to make sure this is an error in the book.
You can't factor this right?

**skeeter** · September 15th 2011, 02:19 PM

Originally Posted by habibixox

3x^2+5x-7

I want to make sure this is an error in the book.
You can't factor this right?

$b^2 - 4ac = 25 - 4(3)(-7) = 109$ ... not a perfect square, therefore will not factor over the rationals.

**habibixox** · September 15th 2011, 02:26 PM

thanks

**Plato** · September 15th 2011, 02:34 PM

Originally Posted by habibixox

3x^2+5x-7
I want to make sure this is an error in the book.
You can't factor this right?

I can factor it,
$\left( {x - \left[ {\frac{{ - 5}}{6} + \frac{{\sqrt {109} }}{6}} \right]} \right)\left( {x - \left[ {\frac{{ - 5}}{6} - \frac{{\sqrt {109} }}{6}} \right]} \right)$

**Prove It** · September 16th 2011, 05:43 AM

Originally Posted by Plato

I can factor it,
$\left( {x - \left[ {\frac{{ - 5}}{6} + \frac{{\sqrt {109} }}{6}} \right]} \right)\left( {x - \left[ {\frac{{ - 5}}{6} - \frac{{\sqrt {109} }}{6}} \right]} \right)$

I certainly hope you mean $\displaystyle 3\left[x - \left(-\frac{5}{6} + \frac{\sqrt{109}}{6}\right)\right]\left[x - \left(-\frac{5}{6} - \frac{\sqrt{109}}{6}\right)\right]$ ...

$\displaystyle \begin{align*} 3x^2 + 5x - 7 &= 3 \left( x^2 + \frac{5}{3}x - \frac{7}{3} \right) \\ &= 3 \left[ x^2 + \frac{5}{3}x + \left( \frac{5}{6} \right)^2 - \left(\frac{5}{6}\right)^2 - \frac{7}{3}\right] \\ &= 3\left[\left(x + \frac{5}{6}\right)^2 - \frac{25}{36} - \frac{84}{36}\right] \\ &= 3\left[\left(x + \frac{5}{6}\right)^2 - \frac{109}{36}\right] \\ &= 3\left[\left(x + \frac{5}{6}\right)^2 - \left(\frac{\sqrt{109}}{6}\right)^2\right] \\ &= 3\left(x + \frac{5}{6} - \frac{\sqrt{109}}{6}\right)\left(x + \frac{5}{6} + \frac{\sqrt{109}}{6}\right) \end{align*}$

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