I am currently struggling to figure out how to solve this problem as I study for an algebra exam. The math problem is:
The two solutions of the quadratic equation $3x^2 - x + k = 0$ are $\frac{p}{4}$ and $p + 1$. Determine the values of $k$ and $p$.
I am not sure how I am supposed to approach the problem, as I have tried every method I could think of. Would be great if anyone can help!
We know the two solutions: $\frac{p}{4}$ and $p+1$
So, just substitute $x$ in the equation with the above values.
$$3(\frac{p}{4})^2 -\frac{p}{4}+k =0$$
and,
$$3(p+1)^2 - (p+1) +k=0$$
Now you have two equations, can you proceed?
Another method:
$\frac{p}{4}+p+1 = \frac{-(-1)}{3}$
and $\frac{p}{4}(p+1)=\frac{k}{3}$
Through Vieta's relations.
In general, if there is a quadratic in form of $ax^2+bx+c=0$, having two roots $\alpha, \beta$, then:
$$\alpha+ \beta= \cfrac{-b}{a}$$
and $$\alpha \beta = \cfrac{c}{a}$$