Solving Quadratic Equations: Find k

[Albert1]

the solutions of :

$x^2+kx+k=0 "

are $ $sin \,\theta \,\,and \,\, cos\, \theta $

please find : $k=?$

 

[soroban]

Hello, Albert!

$$\text{The solutions of: }\: x^2+kx+k\:=\:0$$
$$\text{are }\sin\theta\text{ and }\cos\theta$$

$$\text{Find }k.$$

Spoiler

Since $$k \,=\,\sin\theta\cos\theta$$, we see that: .$$|k| \,<\,1.$$

Quadratic Formula: .$$x \:=\:\frac{-k \pm \sqrt{k^2-4k}}{2}$$

$$\text{Let: }\:\begin{Bmatrix}\sin\theta &=& \frac{-k + \sqrt{k^2-4k}}{2} \\ \cos\theta &=& \frac{-k -\sqrt{k^2-4k}}{2} \end{Bmatrix}$$

$$\text{Then: }\:\begin{Bmatrix}\sin^2\theta &=& \frac{2k^2 - 4k + 2k\sqrt{k^2-4k}}{4} \\ \cos^2\theta &=& \frac{2k^2 - 4k - 2k\sqrt{k^2-4k}}{4} \end{Bmatrix}$$

$$\text{Add: }\:\sin^2\theta + \cos^2\theta \:=\:\frac{4k^2 - 8k}{4} \:=\:1$$

$$\text{And we have: }\:k^2 - 2k - 1\:=\:0$$

$$\text{Hence: }\:k \:=\:1\pm\sqrt{2}$$$$\text{Since }|k| < 1\!:\;k \:=\:1-\sqrt{2}$$

 

[Saitama]

Spoiler

By Vieta's formulas, we have
$$k=\sin\theta \cos\theta$$
$$\sin\theta+\cos\theta=-k$$
Squaring both the sides of second equation,
$$1+2\sin\theta \cos\theta=k^2 \Rightarrow k^2-2k=1 \Rightarrow k^2-2k+1=2 \Rightarrow (k-1)^2=2$$
$$\Rightarrow k=1\pm \sqrt{2}$$
But $|k|<1$, hence, $k=1-\sqrt{2}$.

 

[Albert1]

thanks all for your participation:)

your answers are correct !

 

[CellCoree]

As a scientist, it is important to approach problems and equations with a critical and analytical mindset. In this case, we are given a quadratic equation with unknown variable k and are told that the solutions are sin θ and cos θ. Firstly, it is important to note that the solutions of a quadratic equation are typically in the form of x = a, where a is a numerical value. Therefore, stating that the solutions are sin θ and cos θ is not a typical way of representing the solutions of a quadratic equation.

Furthermore, the given information is not enough to solve for the value of k. We need at least one numerical value or additional equation to determine the value of k. Sin θ and cos θ are trigonometric functions and do not provide enough information to solve for k. Without any additional information, it is not possible to find the value of k.

In conclusion, the given information is insufficient to solve for the value of k in the quadratic equation. As a scientist, it is important to critically analyze and question any given information before attempting to solve a problem. We need more information or a different approach to determine the value of k in this equation.