Solve 3x^2 - 5x - 2 > 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve 3x² − 5x − 2 > 0.

There is an x², so the answer can be two parts.

3x² − 5x − 2 > 0

1 Find two numbers

The number in front of x² is 3.

So we multiply 3 by −2: that is −6.

We need two numbers that multiply to −6 and add to −5.

3 × (−2) = −6

? × ? = −6, ? + ? = −5

They multiply to a minus number, so one number is plus and one is minus.

They add to a minus number, so the bigger one is minus.

× is − → one +, one −; bigger one −

We list the pairs that make 6, smallest first.

We take the smaller from the bigger.

We need 5.

1 and 6 give 5.

Yes!

With the signs: 1 and −6.

1 × 6: 6 − 1 = 5 ✓

2 Split the middle

We write −5x as x − 6x.

It is the same amount.

3x² + x − 6x − 2 > 0

3 Group in pairs

We put the first two together, and the last two together.

(3x² + x) + (−6x − 2) > 0

From the first pair we take out x.

From the second pair we take out −2.

Both leave the same bracket.

(3x² + x) = x(3x + 1)

(−6x − 2) = −2(3x + 1)

Now both parts have 3x + 1 in them.

x(3x + 1) − 2(3x + 1) > 0

4 Same bracket out

3x + 1 is in both parts, so we take it out once.

What is left, x − 2, goes in the other bracket.

ab + cb = b(a + c)

So we get (3x + 1)(x − 2) > 0.

(3x + 1)(x − 2) > 0

5 Find the critical points

A critical point is an x where the side with x is 0, or has no value.

Only there can its sign change.

critical point: where it is 0 (or ÷ 0)

We make each bracket 0.

3x + 1 = 0 x − 2 = 0

In 3x + 1 = 0, we take away 1 on both sides.

In x − 2 = 0, we add 2 on both sides.

3x + 1 − 1 = 0 − 1 x − 2 + 2 = 0 + 2

Now we have 3x = −1 and x = 2.

3x = −1 x = 2

To remove times 3, we divide both sides by 3.

The critical points are −1/3 and 2.

x = −13 x = 2

6 Make a sign table

Between the critical points the sign stays the same.

So one test number tells us the sign of each part.

We draw a table.

The critical points −1/3 and 2 split the number line into 3 parts.

At −1/3, 3x + 1 is 0.

So the product is 0 there.

At 2, x − 2 is 0.

So the product is 0 there.

Left of −1/3, we test x = −1.

−1

3x + 1 is −2 and x − 2 is −3.

−2

−3

Minus times minus is plus.

Between −1/3 and 2, we test x = 0.

3x + 1 is 1 and x − 2 is −2.

−2

Plus times minus is minus.

Right of 2, we test x = 3.

3x + 1 is 10 and x − 2 is 1.

Plus times plus is plus.

7 Read the answer

Greater than 0 means positive.

So we want the parts with a plus.

So x < −1/3 or x > 2.

x < −13 or x > 2

> 0 → the + parts

Careful!

Not the minus part!

Greater than 0 means plus.

Careful: not −13 < x < 2 ✗

8 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A round bracket means the end is not in, like the open circle.

∞ means it goes on for ever.

∪ means or.

interval: (−∞, −13) ∪ (2, ∞)

9 Check

Let's check with a picture: the graph of y = 3x² − 5x − 2.

It is a parabola.

It crosses the x-axis at the critical points, −1/3 and 2.

Above the x-axis, y is plus.

That is x < −1/3 or x > 2: our answer.

Let's check: a number from the answer, 4, put into the question.

x = 4: 3(4)² − 5(4) − 2 = 48 − 20 − 2 = 26

26 is greater than 0.

True!

26 > 0 true

Let's look back.

First: find two numbers.

Then: split the middle.

Then: group in pairs.

Then: same bracket out.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer