Find the range of values of $k$ for which the following equation has real roots.

Question

Find the range of values of $k$ for which the following equation $x^2+(1-k)x=k$ has real roots.

I tried it, for real roots, $b^2-4ac \geqslant 0$

$x^2+(1-k)x-k=0$

$(1-k)^2-4*1*(-k)\geqslant0$

${(1)^2-2\cdot1\cdot k+(k)^2}-4(-k)\geqslant0$

$1-2k+k^2+4k\geqslant0$

$k^2+2k+1\geqslant0$

$k^2+k+k+1\geqslant0$

$(k+1)(k+1)\geqslant0$

Solving the equation I get $k\geqslant -1$. Is this right?

In the book, the answer given is "all values of $k$"

What does that mean?

Answer 1

The discriminant ($\Delta$) of the quadratic equation $x^2 + (1-k)x - k = 0$ is (using the $b^2 - 4ac$ "mnemonic") :

$$\begin{align}\Delta &= (1-k)^2 -4(1)(-k)\\ &=1 - 2k + k^2 + 4k\\ &= 1 + 2k + k^2\\ &= (k + 1)^2\end{align}$$

Now, a quadratic equation has only real roots if and only if $\Delta \ge 0$. This condition is met for all $k$, provided that $k \in \mathbb{R}$ (because the square of any real number is at least $0$).

It will generally fail otherwise (take $k = i = \sqrt{-1}$, the imaginary unit for example). It does not work for all numbers in general, if we were to consider complex and hypercomplex numbers.

So the required values of $k$ is $k\in\mathbb{R}$.

Answer 2

Recall the formula $x_{1,2}=\frac{-b+-\sqrt{b^2-4ac}}{2a}$. This means that $x_{1,2}$ has real values if and only if $b^2-4ac$ is positive (because it is under square root). Now, since you've found that $b^2-4ac=(1+k)(1+k)=(1+k)^2\geqslant0$ it means that for any $k\in\mathbb{R}$ we can find $x_{1,2}$, so it has real roots for any $k\in\mathbb{R}$.