In the equation x^2 - 12x + c = 0, c is a constant. The equation has… Step by Step

Step by step

The question

Question: for which c does x² − 12x + c = 0 have exactly one solution?

x² − 12x + c = 0

We use the discriminant, b² − 4ac.

we want: exactly one solution

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Find a, b and c

We read the numbers.

a = 1 is with x².

b = −12 is with x.

The plain number is the letter c: it is the c we want.

a = 1, b = −12, c = ?

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.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

We write it: b² − 4ac = 0.

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.

(−12)² − 4 × 1 × c = 0

Careful!

(−12)² means (−12) times (−12), not times 2.

Careful: not (−12)² = −24 ✗

(−12) squared is 144.

Minus times minus is plus.

4 times 1 times c is 4c.

(−12)² = (−12) × (−12) = 144

4 × 1 × c = 4c

Now we have 144 − 4c = 0.

144 − 4c = 0

We want c by itself.

First we remove the + 144, so −4c is alone.

Then we remove the times −4.

Is c alone now?

Not yet: + 144 is still with c.

How do we remove it?

4 Remove the 144

To remove 144, we take away 144.

We do it on both sides, so they stay equal.

~~144~~ − 4c ~~− 144~~ = 0 − 144

144 and − 144 cancel each other.

They make zero.

0 − 144 = −144.

0 − 144 = −144

Now we have −4c = −144.

−4c = −144

So after step 4: we took away 144, and now we have −4c = −144.

← step 4: took away 144

Is c alone now?

Not yet: −4c means −4 times c.

5 Remove the −4

To remove times −4, we divide by −4.

We do it on both sides, so they stay equal.

−4c−4 = −144−4

−4 divided by −4 is 1.

They cancel each other.

So just c is left.

−144 divided by −4 is 36.

−144−4 = 36

So c = 36!

c = 36

6 Check

Let's check.

We put c = 36 back in.

x² − 12x + 36 = 0

b² − 4ac is 144 take away 144: 0.

b² − 4ac = (−12)² − 4 × 1 × 36 = 144 − 144 = 0

In the quadratic formula, the square root of 0 is 0.

Plus 0 and minus 0 give the same number: x = 6.

Exactly one solution.

It works!

x = 12 ± 02 = 6 only

7 Pick the choice

Our answer is c = 36.

That is choice C.

C) 36

8 Why not the others

A is −6.

Wrong: half of −12, but we still had to square it.

A) −6 ✗ half of −12, but we still had to square it

B is 6.

Wrong: half of 12, and we still had to square it.

B) 6 ✗ half of 12, and we still had to square it

D is 144.

Wrong: 12 × 12: we square half of 12, not 12.

D) 144 ✗ 12 × 12: we square half of 12, not 12

Let's look back.

First: find a, b and c.

Then: choose the rule.

Then: put the numbers in.

Then: remove the 144.

Then: remove the −4.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer