How do you solve the following with the quadratic formula?: sqrt2x² - x - 3sqrt(2) =0

Jul 8, 2018

$x = \frac{3 \sqrt{2}}{2} , \mathmr{and} , x = - \sqrt{2}$.

Explanation:

Comparing the given quadratic equation with the standard one, i.e.,

$a {x}^{2} + b x + c = 0$, we have,

$a = \sqrt{2} , b = - 1 , \mathmr{and} c = - 3 \sqrt{2}$.

As per the quadratic formula, the roots are,

$x = \frac{- a \pm \sqrt{\Delta}}{2 a} , \text{ where, } \Delta = \left({b}^{2} - 4 a c\right)$.

In our case, $\Delta = {\left(- 1\right)}^{2} - 4 \left(\sqrt{2}\right) \left(- 3 \sqrt{2}\right)$,

$= 1 + 24$.

$\therefore \Delta = 25 , \text{ so that, } \sqrt{\Delta} = 5$.

Hence, $- b \pm \sqrt{\Delta} = 1 \pm 5 = 6 , \mathmr{and} , - 4$.

Finally, we get the roots : $\frac{6}{2 \sqrt{2}} , \mathmr{and} , - \frac{4}{2 \sqrt{2}} ,$

$i . e . , \frac{3}{\sqrt{2}} = \frac{3 \sqrt{2}}{2} , \mathmr{and} , - \sqrt{2}$.

Jul 8, 2018

$x = - \sqrt{2} , \frac{3 \sqrt{2}}{2}$

Explanation:

Given: $\sqrt{2} {x}^{2} - x - 3 \sqrt{2} = 0$.

Use the quadratic formula, which states that,

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Here, $a = \sqrt{2} , b = - 1 , c = - 3 \sqrt{2}$.

$\therefore x = \frac{1 \pm \sqrt{1 - 4 \cdot \sqrt{2} \cdot - 3 \sqrt{2}}}{2 \sqrt{2}}$

$= \frac{1 \pm \sqrt{1 + 24}}{2 \sqrt{2}}$

$= \frac{1 \pm \sqrt{25}}{2 \sqrt{2}}$

$= \frac{1 \pm 5}{2 \sqrt{2}}$

$\therefore {x}_{1} = \frac{1 + 5}{2 \sqrt{2}} = \frac{6}{2 \sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3 \sqrt{2}}{2}$

$\therefore {x}_{2} = \frac{1 - 5}{2 \sqrt{2}} = - \frac{4}{2 \sqrt{2}} = - \frac{2}{\sqrt{2}} = - \sqrt{2}$