How many solutions 3x^2 - 4x - 2 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of 3x² − 4x − 2 = 0.

3x² − 4x − 2 = 0

1 Find a, b and c

We read the numbers: a = 3 is with x², b = −4 is with x, c = −2 is the plain number.

a = 3, b = −4, c = −2

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 = (−4)² − 4 × 3 × (−2)

(−4)² is 16.

Minus times minus is plus.

4 times 3 times −2 is −24.

(−4)² = (−4) × (−4) = 16

4 × 3 × (−2) = −24

Careful!

b = −4 goes in brackets.

(−4) squared is 16, not minus 16.

Careful: not −4² = −16 ✗

So we have 16 take away −24.

= 16 − (−24)

Taking away a negative is the same as adding.

16 − (−24) = 16 + 24

So the discriminant is 40.

= 40

3 How many solutions

40 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 3x² − 4x − 2 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 40.

x = 4 ± √406

With a calculator: about 1.7 and about −0.4.

Two different solutions.

It works!

x ≈ 1.7 or x ≈ −0.4

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer