Step by step
The question
Question: for which k does x² + kx + 25 = 0 have exactly one solution?
1 Find a, b and c
We read the numbers.
a = 1 is with x².
With x is the letter k.
c = 25 is the plain number.
2 Choose the rule
From the discriminant lesson: the sign of b² − 4ac tells how many solutions there are.
We want exactly one solution.
So b² − 4ac must be 0.
We write it: b² − 4ac = 0.
3 Put the numbers in
We put in the numbers.
The letter k stays as it is: it is the number we want.
4 times 1 times 25 is 100.
Now we have k² − 100 = 0.
4 Remove the − 100
We want k² alone.
− 100 is in the way.
So we add 100 on both sides.
We add 100 on both sides.
They stay equal.
− 100 and + 100 make zero.
Cross them out.
Now we have k² = 100.
5 Find k
Which numbers times themselves make 100?
10 times 10 is 100.
And −10 times −10 is 100 too: minus times minus is plus.
Careful!
k = −10 works too.
(−10)² is 100.
So k = 10 or k = −10.
6 Check
Let's check.
k = 10: 10 squared is 100, and 100 − 100 is 0.
And k = −10: (−10) squared is 100, and 100 − 100 is 0.
Both give exactly one solution.
It works!
Let's look back.
First: find a, b and c.
Then: choose the rule.
Then: put the numbers in.
Then: remove the − 100.
Then: find k.
Last: check.