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

Step by step

The question

Question: solve −x² + 2x + 3 > 0.

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

−x² + 2x + 3 > 0

We want x² to be plus.

So we times both sides by −1.

Times a negative number turns the sign around.

Greater than becomes less than.

x² − 2x − 3 < 0

1 Find two numbers

We need two numbers.

They multiply to −3, and they add to −2.

? × ? = −3

? + ? = −2

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 3, smallest first.

We take the smaller from the bigger.

We need 2.

1 and 3 give 2.

Yes!

With the signs: 1 and −3.

1 × 3: 3 − 1 = 2 ✓

The two numbers go in the brackets.

So we get (x + 1)(x − 3) < 0.

(x + 1)(x − 3) < 0

2 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.

x + 1 = 0 x − 3 = 0

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

In x − 3 = 0, we add 3 on both sides.

x + 1 − 1 = 0 − 1 x − 3 + 3 = 0 + 3

The critical points are −1 and 3.

x = −1 x = 3

3 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 and 3 split the number line into 3 parts.

At −1, x + 1 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of −1, we test x = −2.

−2

x + 1 is −1 and x − 3 is −5.

−1

−5

Minus times minus is plus.

Between −1 and 3, we test x = 0.

x + 1 is 1 and x − 3 is −3.

−3

Plus times minus is minus.

Right of 3, we test x = 4.

x + 1 is 5 and x − 3 is 1.

Plus times plus is plus.

4 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So −1 < x < 3.

−1 < x < 3

< 0 → the − parts

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x < −1 or x > 3 ✗

5 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.

interval: (−1, 3)

6 Check

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

It is a parabola.

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

Below the x-axis, y is minus.

That is −1 < x < 3: our answer.

Let's check: a number from the answer, 0, put into the question.

x = 0: −(0)² + 2(0) + 3 = 0 + 0 + 3 = 3

3 is greater than 0.

True!

3 > 0 true

Let's look back.

First: find two numbers.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer