solve the equation x^2 - (6√3)x^2 + 24 = 0 giving the answer as surds, simplified as far as possible.
i can't isolate the '6√3' without getting something wrong and ruining the equation.
solve the equation x^2 - (6√3)x^2 + 24 = 0 giving the answer as surds, simplified as far as possible.
i can't isolate the '6√3' without getting something wrong and ruining the equation.
If you didn't mean to square the $\displaystyle x$ in the second term of the equation;
i.e. $\displaystyle x^2-(6\sqrt{3})x+24=0$
Then you could either use the quadratic formula or complete the square (as follows) to solve for $\displaystyle x$
$\displaystyle x^2-(6\sqrt{3})x+24=0$
$\displaystyle \left (x-\frac{6\sqrt{3}}{2}\right ) ^2-\frac{108}{4}+\frac{96}{4}=0$
$\displaystyle \left ( x-\frac{6\sqrt{3}}{2}+\frac{2\sqrt{3}}{2}\right )\left ( x-\frac{6\sqrt{3}}{2}-\frac{2\sqrt{3}}{2}\right )=0$
$\displaystyle (x-2\sqrt{3})(x-4\sqrt{3})=0$
So, $\displaystyle x=2\sqrt{3}$ or $\displaystyle x=4\sqrt{3}$
Hope that helps
If the equation is $\displaystyle x^2-(6\sqrt{3})x^2+24=0$
Then you can rearrange it to the form:
$\displaystyle (1-6\sqrt{3})x^2+24=0$ ,as when you expand $\displaystyle (1-6\sqrt{3})x^2$ you get $\displaystyle x^2-(6\sqrt{3})x^2$
From there you can simply solve as follows:
$\displaystyle (1-6\sqrt{3})x^2+24=0$
$\displaystyle (1-6\sqrt{3})x^2=-24$
$\displaystyle x^2=\frac{-24}{1-6\sqrt{3}}$
$\displaystyle x=\pm\sqrt{\frac{24}{6\sqrt{3}-1}}$
You should have a look here: Practical Algebra Lessons under the "intermediate algebra topics"; it explains quadratics a lot better than I'm able to.
For the method I used (completing the square) go to Completing the Square: Solving Quadratic Equations otherwise there's also a link there on how to use the quadratic formula - which is essentially the same thing as it's derived by completing the square - where you can just plug in values into the formula to solve for $\displaystyle x$
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