Step by step
The question
Question: for which k does x² + kx + 9 = 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 = 9 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 9 is 36.
Now we have k² − 36 = 0.
4 Remove the − 36
We want k² alone.
− 36 is in the way.
So we add 36 on both sides.
We add 36 on both sides.
They stay equal.
− 36 and + 36 make zero.
Cross them out.
Now we have k² = 36.
5 Find k
Which numbers times themselves make 36?
6 times 6 is 36.
And −6 times −6 is 36 too: minus times minus is plus.
Careful!
k = −6 works too.
(−6)² is 36.
So k = 6 or k = −6.
6 Check
Let's check.
k = 6: 6 squared is 36, and 36 − 36 is 0.
And k = −6: (−6) squared is 36, and 36 − 36 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 − 36.
Then: find k.
Last: check.