Step by step
The question
Question: for which c does x² − 10x + c = 0 have exactly one solution?
1 Find a, b and c
We read the numbers.
a = 1 is with x².
b = −10 is with x.
The plain number is the letter c: it is the c we want.
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, each negative one in brackets.
The letter c stays as it is: it is the number we want.
Careful!
(−10)² means (−10) times (−10), not times 2.
(−10) squared is 100.
Minus times minus is plus.
4 times 1 times c is 4c.
Now we have 100 − 4c = 0.
We want c by itself.
Two things are stuck to c: times −4 and + 100.
First c was times −4.
Then + 100.
To take them off, we go backwards.
First we remove the + 100.
Then the times −4.
4 Remove the 100
To remove 100, we take away 100.
We do it on both sides, so they stay equal.
100 and − 100 make zero.
Cross them out.
0 − 100 = −100.
Now we have −4c = −100.
5 Remove the −4
−4c means −4 times c.
We want just c.
To remove times −4, we divide by −4.
We do it on both sides, so they stay equal.
−4 divided by −4 is 1.
So just c is left.
−100 divided by −4 is 25.
So c = 25!
6 Check
Let's check.
We put c = 25 back in.
b² − 4ac is 100 take away 100: 0.
In the quadratic formula, the square root of 0 is 0.
Plus 0 and minus 0 give the same number: x = 5.
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: remove the −4.
Last: check.