Discriminant of 4x^2 - 20x + 25 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of 4x² − 20x + 25 = 0.

4x² − 20x + 25 = 0

1 Find a, b and c

We read the numbers: a = 4 is with x², b = −20 is with x, c = 25 is the plain number.

a = 4, b = −20, c = 25

2 Work out b² − 4ac

In the formula, b² − 4ac is under the square root.

It is called the discriminant.

Its sign tells us how many solutions there are.

x = −b ± √b² − 4ac2a

We put the numbers in, each negative one in brackets.

b² − 4ac = (−20)² − 4 × 4 × 25

(−20)² is 400.

Minus times minus is plus.

4 times 4 times 25 is 400.

(−20)² = (−20) × (−20) = 400

4 × 4 × 25 = 400

Careful!

b = −20 goes in brackets.

(−20) squared is 400, not minus 400.

Careful: not −20² = −400 ✗

So we have 400 take away 400.

= 400 − 400

So the discriminant is 0.

= 0

3 How many solutions

It is 0.

The square root of 0 is 0.

Plus 0 and minus 0 give the same answer.

So there is just one.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So 4x² − 20x + 25 = 0 has 1 real solution.

= 1 real solution

4 Check

Let's check with the formula.

Under the root we get 0, and the square root of 0 is 0.

x = 20 ± √08 = 20 ± 08

Plus 0 and minus 0 give the same number: x = 5/2.

One solution.

It works!

x = 52 only

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer