# Find :(b^2/2a)+(a^2/2b)

#### Albert

##### Well-known member
$a^2=3a+5$
$b^2=3b+5$
$a\neq b$
$\dfrac {b^2}{2a}+\dfrac {a^2}{2b}=?$

#### anemone

##### MHB POTW Director
Staff member
Re: find b^2/2a)+(a^2/2b)

Since we're given two quadratic equations $$\displaystyle a^2=3a+5$$ and $$\displaystyle b^2=3b+5$$ and that $$\displaystyle a\ne b$$, we can tell by quadratic formula that

$$\displaystyle a=\frac{3+ \sqrt{29}}{2}$$ and $$\displaystyle b=\frac{3- \sqrt{29}}{2}$$.

Thus, $$\displaystyle a+b=3$$ and $$\displaystyle ab=-5$$.

Therefore,

$$\displaystyle \frac{b^2}{2a}+\frac{a^2}{2b}$$

$$\displaystyle =\frac{b^3}{2ab}+\frac{a^3}{2ab}$$

$$\displaystyle =\frac{a^3+b^3}{2ab}$$

$$\displaystyle =\frac{(a+b)(a^2+b^2-ab)}{2ab}$$

$$\displaystyle =\frac{(3)(3(3)+10-(-5))}{2(-5)}$$

$$\displaystyle =-7.2$$

P.S. The values for a and b are interchangeable.