I must have forgotten some rule... trying to figure out how to simplify this...
\(x=\frac{4\left(a+b\right)±\sqrt{16\left(a+b\right)^2-48ab}}{24}\)
Simplifies to:
\(x=\frac{\left(a+b\right)±\sqrt{a^2-ab+b^2}}{6}\)
Can someone please explain the rules and steps being used here for this simplification?
If it helps the function is:
f'(x)=12x2 - 4(a+b)x + ab
Thanks!
NVM figured it out...
\(x=\frac{4\left(a+b\right)+\sqrt{16\left(a^2+2ab+b^2\right)-48ab}}{24}\)
\(x=\frac{4\left(a+b\right)+\sqrt{16\left(a^2+2ab+b^2-3ab\right)}}{24}\)
\(x=\frac{4\left(a+b\right)+\sqrt{16\left(a^2-ab+b^2\right)}}{24}\)
\(x=\frac{4\left(a+b\right)+\sqrt{16}\sqrt{a^2-ab+b^2}}{24}\)
\(x=\frac{4\left(a+b\right)+4\sqrt{a^2-ab+b^2}}{24}\)
\(x=\frac{\left(a+b\right)+\sqrt{a^2-ab+b^2}}{6}\)
+26387 I must have forgotten some rule... trying to figure out how to simplify this...
\(x=\frac{4\left(a+b\right)±\sqrt{16\left(a+b\right)^2-48ab}}{24}\)
Simplifies to: \(x=\frac{\left(a+b\right)±\sqrt{a^2-ab+b^2}}{6}\)
\(\begin{array}{|rcll|} \hline x &=& \frac{4\left(a+b\right) \pm \sqrt{16\cdot \left(a+b\right)^2-48ab}}{24} \\ &=& \frac{4\left(a+b\right) \pm \sqrt{16\cdot \left(a+b\right)^2-16\cdot 3 \cdot ab}}{24} \\ &=& \frac{4\left(a+b\right) \pm \sqrt{ 16 \cdot[~ \left(a+b\right)^2-3\cdot ab ~]}}{24} \quad & | \quad \sqrt{a\cdot b} = \sqrt{a}\cdot \sqrt{ b} \\ &=& \frac{4\left(a+b\right) \pm \sqrt{ 16}\cdot \sqrt{ \left(a+b\right)^2-3ab } }{24} \quad & | \quad \sqrt{ 16}=4\\ &=& \frac{4\left(a+b\right) \pm 4\cdot \sqrt{ \left(a+b\right)^2-3ab } }{24} \\ &=& \frac{4 \cdot[~ \left(a+b\right) \pm \sqrt{ \left(a+b\right)^2-3ab} ~] }{24} \\ &=& \frac{4 \cdot[~ \left(a+b\right) \pm \sqrt{ \left(a+b\right)^2-3ab} ~] }{4\cdot 6} \\ &=& \frac{ \left(a+b\right) \pm \sqrt{ \left(a+b\right)^2-3ab} } { 6} \quad & | \quad (a+b)^2 = a^2 + 2ab + b^2\\ &=& \frac{ \left(a+b\right) \pm \sqrt{ a^2 + 2ab + b^2-3ab} } { 6} \\ &=& \frac{ \left(a+b\right) \pm \sqrt{ a^2 + 2ab-3ab + b^2} } { 6} \quad & | \quad 2ab-3ab = ab(2-3)= ab(-1) = -ab\\ &=& \frac{ \left(a+b\right) \pm \sqrt{ a^2 -ab + b^2} } { 6} \\ \hline \end{array}\)
