How many solutions x^2 - 5x + 4 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of x² − 5x + 4 = 0.

x² − 5x + 4 = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = −5 is with x, c = 4 is the plain number.

a = 1, b = −5, c = 4

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 = (−5)² − 4 × 1 × 4

(−5)² is 25.

Minus times minus is plus.

4 times 1 times 4 is 16.

(−5)² = (−5) × (−5) = 25

4 × 1 × 4 = 16

Careful!

b = −5 goes in brackets.

(−5) squared is 25, not minus 25.

Careful: not −5² = −25 ✗

So we have 25 take away 16.

= 25 − 16

So the discriminant is 9.

= 9

3 How many solutions

9 is positive.

The square root of a positive number is a real number.

The plus or minus gives two different answers.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So x² − 5x + 4 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 9.

Its square root is 3.

x = 5 ± 32

The plus gives 4, the minus gives 1.

Two different solutions.

It works!

x = 4 or 1

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer