# Thread: Showing that the equation has real roots for all real values of 'k'

1. ## Showing that the equation has real roots for all real values of 'k'

Hello everyone here on Math Help Forum.

I have an urgent question here: show that the equation $\displaystyle x^2 + kx = 4 - 2k$ has real roots for all real values of $\displaystyle k$.

What I know is that the term ''real roots'' imply that $\displaystyle b^2 - 4ac$ is equal or more than $\displaystyle 0$. But my primal problem is that I have difficulties in showing and proving. So here are my workings:

$\displaystyle x^2 + kx = 4 - 2k$
$\displaystyle x^2 + kx -4 +2k = 0$

Here $\displaystyle a = 1, b = k, c = -4 + 2k$

$\displaystyle b^2 -4ac = k^2 - 4(1)(-4+2k)$
$\displaystyle k^2 -8k +24$
$\displaystyle (x-4)(x-4)$
$\displaystyle (x-4)^2$

I have tried my best. Please pinpoint my mistakes and kindly continue the rest of the question. Thank you so much for your help and have a nice day!

2. $\displaystyle x^2 + kx - 4 + 2k =0$

$\displaystyle a=1, b=k, c= - 4 + 2k$

Now solve for the discriminant >0

3. Originally Posted by PythagorasNeophyte
Hello everyone here on Math Help Forum.

I have an urgent question here: show that the equation $\displaystyle x^2 + kx = 4 - 2k$has real roots for all real values of $\displaystyle k$.

What I know is that the term ''real roots'' imply that $\displaystyle b^2 - 4ac$ is equal or more than $\displaystyle 0$. But my primal problem is that I have difficulties in showing and proving. So here are my workings:

$\displaystyle x^2 + kx = 4 - 2k$
$\displaystyle x^2 + kx -4 +2k = 0$

Here $\displaystyle a = 1, b = k, c = -4 + 2k$

$\displaystyle b^2 -4ac = k^2 - 4(1)(-4+2k)$
$\displaystyle k^2 -8k +24$
$\displaystyle (x-4)(x-4)$
$\displaystyle (x-4)^2$

I have tried my best. Please pinpoint my mistakes and continue the rest of the question. Thank you so much for your help and have a nice day!
1. $\displaystyle k^2-8k+24\ne (x-4)^2$

2. According to the question you are asked to show that

$\displaystyle k^2-8k+24\geq 0 ~for\ all ~ k \in \mathbb{R}$

$\displaystyle k^2-8k+16+8\geq 0 ~for\ all ~ k \in \mathbb{R}$

$\displaystyle (k-4)^2+8\geq 0 ~for\ all ~ k \in \mathbb{R}$

3. A square is allways greater or equal zero. If you add a positive number to a square the sum must be greater than zero.

4. Thank you so much for your replies! I am very sorry for the mistake when I should have written $\displaystyle k$ instead of $\displaystyle x$ when I factorise.

But why $\displaystyle k^2-8k+24$, when factorised, gives $\displaystyle (k-4)^2 + 8$? If I were to factorise, it will be $\displaystyle (k-4)(k-4)$; expanding $\displaystyle (k-4)(k-4)$ gives $\displaystyle k^2-8k+24$.

The most important thing here is to prove that $\displaystyle x^2+kx = 4 -2k$ has real roots for all values of k.

5. Originally Posted by PythagorasNeophyte
Thank you so much for your replies! I am very sorry for the mistake when I should have written $\displaystyle k$ instead of $\displaystyle x$ when I factorise.

But why $\displaystyle k^2-8k+24$, when factorised, gives $\displaystyle (k-4)^2 + 8$? If I were to factorise, it will be $\displaystyle (k-4)(k-4)$; expanding $\displaystyle (k-4)(k-4)$ gives $\displaystyle k^2-8k+24$.
Never!
$\displaystyle (k-4)(k-4) = k^2-4k-4k+(-4)(-4)$

And I assure you $\displaystyle 4 \cdot 4 \ne 24$
The most important thing here is to prove that $\displaystyle x^2+kx = 4 -2k$ has real roots for all values of k.
If the discriminant - that's what you have calculated - is greater or equal to zero then there exist real roots. Compare your own text at the beginning.

6. Oh yes, I have forgotten that I am solving an inequality. I can only factorise in that way when I am solving an equation. I am very sorry for that.

$\displaystyle k^2 -8k +24 \geq 0$

$\displaystyle \Rightarrow (k-4)^2 + 8 \geq 0$

What do I do with the 8?

7. Originally Posted by pickslides
$\displaystyle x^2 + kx - 4 + 2k =0$

$\displaystyle a=1, b=k, c= - 4 + 2k$

Now solve for the discriminant >0
Using what I have already said before hand

$\displaystyle \displaystyle b^2-4ac > 0 \implies k^2 -4(1)(-4+2k)>0$

$\displaystyle \displaystyle k^2 -4(1)(-4+2k)>0$

$\displaystyle \displaystyle k^2 -4(-4+2k)>0$

$\displaystyle \displaystyle k^2 +16-8k>0$

$\displaystyle \displaystyle (k-4)^2>0$

$\displaystyle \dots$

8. Originally Posted by pickslides
Using what I have already said before hand

$\displaystyle \displaystyle b^2-4ac > 0 \implies k^2 -4(1)(-4+2k)>0$

$\displaystyle \displaystyle k^2 -4(1)(-4+2k)>0$

$\displaystyle \displaystyle k^2 -4(-4+2k)>0$

$\displaystyle \displaystyle k^2 +16-8k>0$

$\displaystyle \displaystyle (k-4)^2>0$

$\displaystyle \dots$
You are using strict inequalities when you should be using $\displaystyle $$\ge (when the discriminant is zero the original polynomial has a real double root). The discriminant has a double root at \displaystyle$$k=4$ and since the leading coefficient is positive the discriminant is always non-negative.

CB

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# show that the roots of the equation are real for all real values of M

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