1

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!

Rodney Millan
  • 25

1 Answers1

2

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 $$\alpha \beta = \cfrac{c}{a}$$

  • Thank you so much, I do not know how I didn't think of this... Would you be able to make a similar question so I can solidify this knowledge? – Rodney Millan May 28 '21 at 06:30
  • Sure. If $-4$ is a root of the quadratic equation $x^2+px-4=0$, and the quadratic equation $x^2+px+k=0$ has equal roots, find the value of $p$ and $k$. –  May 28 '21 at 06:36
  • This question involves the concept of equal roots, do you know about this? If not, I can change the question as per your current knowledge. –  May 28 '21 at 06:37
  • I am not aware of the concept, is it difficult to learn? – Rodney Millan May 28 '21 at 06:40
  • @RodneyMillan, Not at all. You will get it once you learn the quadratic formula: $$\cfrac{-b\pm \sqrt{b^2-4ac}}{2a}$$ –  May 28 '21 at 06:45
  • @RodneyMillan, refer to this pdf for Vieta's relations (For more details) https://www.andrew.cmu.edu/user/daltizio/Vietas%20Formulas.pdf –  Jun 02 '21 at 12:06