My professor insists these quadratic equations can be factored, she’s asking us to graph them as well. Maybe its just me...
$\displaystyle 2x^2-3x-5=0$
&
$\displaystyle x^2-4x+2=0$
Any suggestions?
My professor insists these quadratic equations can be factored, she’s asking us to graph them as well. Maybe its just me...
$\displaystyle 2x^2-3x-5=0$
&
$\displaystyle x^2-4x+2=0$
Any suggestions?
To find out if a quadratic can be factored calculate the discriminant, $\displaystyle \Delta$.
$\displaystyle \Delta = b^2-4ac$
If $\displaystyle \Delta$ is a perfect square it can be factorised.
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$\displaystyle (-3)^2-(4)(2)(-5) = 49$ ------- This can be factored
$\displaystyle (-4)^2-(4)(1)(2) = 8$ --------- This cannot be factored
Any quadratic "can" be factored, just not necessarily with integer coefficients.
For $\displaystyle 2x^2- 3x- 5$, by the quadratic formula, its zeros are $\displaystyle \frac{3\pm\sqrt{9+ 40}}{6}= \frac{3\pm 7}{4}$ or 5/2 and -1. $\displaystyle 2x^2- 3x- 5= 2(x- 5/2)(x+ 1)= (2x- 5)(x+ 1)$
For $\displaystyle x^2- 4x+ 2$ we can complete the square: $\displaystyle x^2- 4x+ 2= x^2- 4x+ 4- 4+ 2= (x- 2)^2- 2= (x-2+ \sqrt{2})(x-2-\sqrt{2})$. the last step is factoring the "difference of two squares", (x-2)^2 and $\displaystyle (\sqrt{2})^2$