Sum and product of the roots of 2x^2 + 3x

Step by step

The question

Question: find the sum and the product of the roots of 2x² + 3x − 2 = 0.

2x² + 3x − 2 = 0

The roots are the two answers of the equation.

We call them r and s.

1 Read a, b and c

In a x squared plus b x plus c: a = 2, b = 3, c = −2.

= a = 2, b = 3, c = −2

Why is there a rule?

With roots r and s, the equation is x minus r, times x minus s.

Opened, that is x squared, minus r plus s times x, plus r s.

(x − r)(x − s) = x² − (r + s)x + rs

So the sum is minus b over a.

The product is c over a.

With a in front, we divide by a first.

so: r + s = −ba, rs = ca

2 Sum of the roots

The sum is minus b/a: minus 3, over 2.

That is −3/2.

r + s = −32 = −32

Careful!

The sum has a minus: minus b over a.

3/2 has the wrong sign.

Careful: not r + s = 32 ✗

3 Product of the roots

The product is c/a: −2/2.

That is −1.

rs = −22 = −1

So the sum of the roots is −3/2 and the product is −1.

r + s = −32, rs = −1

4 Check

Let's check.

Solving the equation gives x = 1/2 and x = −2.

the roots: x = 12 and x = −2

Their sum is −3/2 and their product is −1.

The same!

So the answer is right.

sum: 12 + (−2) = −32 product: 12 × (−2) = −1

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer