Sum and product of the roots of x^2 - 2x

Step by step

The question

Question: find the sum and the product of the roots of x² − 2x − 8 = 0.

x² − 2x − 8 = 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 = 1, b = −2, c = −8.

= a = 1, b = −2, c = −8

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 (−2), over 1.

That is 2.

r + s = −(−2) = 2

Careful!

The sum has a minus: minus b over a.

−2/1 has the wrong sign.

Careful: not r + s = −21 ✗

3 Product of the roots

The product is c/a: −8/1.

That is −8.

rs = −8

So the sum of the roots is 2 and the product is −8.

r + s = 2, rs = −8

4 Check

Let's check.

Solving the equation gives x = 4 and x = −2.

the roots: x = 4 and x = −2

Their sum is 2 and their product is −8.

The same!

So the answer is right.

sum: 4 + (−2) = 2 product: 4 × (−2) = −8

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer