Determine the of p and other roots.

Question

One of the roots of $3x^2 + p =5x$, is $2$. Determine the value of $p$ and the other root.

Answer 1

As $2$ is a root of $3x^2-5x+p=0, 3\cdot2^2-5\cdot2+p=0\implies p=-2$

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Alternatively, $$3x^2-5x+p=0$$

If the other root is $b,$ $ 2+b=\frac53\implies b=\frac53-2=-\frac13$ and $ 2\cdot b=\frac p3\implies p=6b=6(-\frac13)=-2$

Answer 2

Hints:

(1) $\,3x^2+p=5x\iff 3x^2-5x+p=0\,$

(2) If we have the quadratic $\,ax^2+bx+c=0\;,\;\;a\neq 0\,$ , then its roots are given by

$$x_{1,2}=\frac{-b\pm\sqrt\Delta}{2a}\;,\;\;\;\Delta:=b^2-4ac$$

(3) The above quadratic has at least one real root iff $\,\Delta\ge 0\,$

Answer 3

Let $x_1=2$ then from Vieta $$2+x_2=5/3$$and$$2x_2=p/3$$ follow that $$x_2=5/3-2=-1/3$$and $$2\cdot(-1/3)=p/3\Rightarrow p=-2$$