In the equation x^2 + 14x + c = 0, c is a constant. The equation has… Step by Step

Step by step

The question

Question: for which c does x² + 14x + c = 0 have exactly one solution?

x² + 14x + 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 = 14 is with x.

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

a = 1, b = 14, 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.

The letter c stays as it is: it is the number we want.

14² − 4 × 1 × c = 0

Careful!

14² means 14 times 14, not times 2.

Careful: not 14² = 28 ✗

14 squared is 196.

4 times 1 times c is 4c.

14² = 14 × 14 = 196

4 × 1 × c = 4c

Now we have 196 − 4c = 0.

196 − 4c = 0

We want c by itself.

First we remove the + 196, so −4c is alone.

Then we remove the times −4.

Is c alone now?

Not yet: + 196 is still with c.

How do we remove it?

4 Remove the 196

To remove 196, we take away 196.

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

~~196~~ − 4c ~~− 196~~ = 0 − 196

196 and − 196 cancel each other.

They make zero.

0 − 196 = −196.

0 − 196 = −196

Now we have −4c = −196.

−4c = −196

So after step 4: we took away 196, and now we have −4c = −196.

← step 4: took away 196

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 = −196−4

−4 divided by −4 is 1.

They cancel each other.

So just c is left.

−196 divided by −4 is 49.

−196−4 = 49

So c = 49!

c = 49

6 Check

Let's check.

We put c = 49 back in.

x² + 14x + 49 = 0

b² − 4ac is 196 take away 196: 0.

b² − 4ac = 14² − 4 × 1 × 49 = 196 − 196 = 0

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

Plus 0 and minus 0 give the same number: x = −7.

Exactly one solution.

It works!

x = −14 ± 02 = −7 only

7 Pick the choice

Our answer is c = 49.

That is choice C.

C) 49

8 Why not the others

A is 7.

Wrong: half of 14, but we still had to square it.

That is the Careful mistake from before.

A) 7 ✗ half of 14, but we still had to square it

B is 14.

Wrong: took 14 itself: c is half of 14, squared.

B) 14 ✗ took 14 itself: c is half of 14, squared

D is 196.

Wrong: 14 × 14: we square half of 14, not 14.

D) 196 ✗ 14 × 14: we square half of 14, not 14

Let's look back.

First: find a, b and c.

Then: choose the rule.

Then: put the numbers in.

Then: remove the 196.

Then: remove the −4.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer