Review question

# When does $x^2-4x-1=2k(x-5)$ have equal roots? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5316

## Solution

The quadratic equation $x^2-4x-1=2k(x-5)$ has two equal roots. Calculate the possible values of $k$.

We can rewrite $x^2-4x-1=2k(x-5)$ as $x^2-(2k+4)x+10k-1=0.$ This has two equal roots exactly when the discriminant of the equation is zero, that is, when $(2k+4)^2-4(10k-1)=0.$ Multiplying out the brackets, we find that $4k^2+16k+16-40k+4=0$ which simplifies to $4k^2-24k+20=0.$ Dividing through by $4$, we find $k^2-6k+5=0$ which factorises to $(k-5)(k-1)=0.$ Therefore the equation has two equal roots exactly when $k=1$ or $k=5$.

To visualise this, you could create a GeoGebra file to show this curve and line, with a slider for $k$. Use this file to confirm our results above.