Hello i need some help factorising the following quadratic equation in order to solve it, i have a few to do but i will post an example, then if working out is shown i should be able to do the rest,
2xsquared -5x - 7 = 0
Thanks....
Hello,
I assume that you are familiar with the 3 binomial formulae which are used here:
$\displaystyle 2x^2-5x-7 = 0~\iff~2(x^2-\frac52 x - \frac72)=0$
$\displaystyle 2(x^2-\frac52 x - \frac72)=0~\iff~ 2\left(x^2-\frac52 x+\frac{25}{16}-\frac{25}{16}-\frac72\right)=0$
$\displaystyle
2\left(\left(x-\frac54\right)^2-\frac{81}{16}\right)=0~\iff~2\left(x-\frac54 + \frac94\right) \cdot \left(x-\frac54-\frac94\right)=0$
Now use the property: A product equals zero if one of the factors equals zero. Thus:
$\displaystyle \left(x+1\right)=0 ~\vee~\left(x-\frac72\right) =0$
And now solve for x.
unfortunately there's no, that is of corse if you don't know the roots of the polynomial beforehand.easier way than this
Actually I don't know why do you insist on factorizing the polynomial, when a simple formula already exists for findingg the roots of a quadratic equation.
$\displaystyle \begin{array}{l}
p(x) = ax^2 + bx + c \\
x_{1,2} = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
\end{array}$
$\displaystyle p(x) = a(x - x_1 )(x - x_2 )$
you might also read about:
Vieta's Formulas -- from Wolfram MathWorld
$\displaystyle 2x^2 - 5x - 7 = 0$
Here's another method that works, but only does so if the roots of the equation are rational numbers or integers.
This is called the "ac" method.
Multiply the leading coefficient by the constant coefficient. In this case $\displaystyle 2 \cdot -7 = -14$
Now write all the pairs of factors of -14:
1, -14
2, -7
7. -2
14, -1
Now look for a pair that sums to the coefficient of the linear term, in this case -5. Note that 2 + (-7) = -5. (If such a pair does not exist, then the equation cannot be factored in terms of rational numbers.)
So we want to write the linear term as
[tex]-5x = (2 - 7)x = 2x - 7x
in the original quadratic.
Thus
$\displaystyle 2x^2 - 5x - 7 = 0$
$\displaystyle 2x^2 + 2x - 7x - 7 = 0$
Now factor by grouping:
$\displaystyle (2x^2 + 2x) + (-7x - 7) = 0$
$\displaystyle 2x(x + 1) + (-7)(x + 1) = 0$
$\displaystyle (2x - 7)(x + 1) = 0$
Now you can solve this by setting each factor independently equal to 0.
-Dan