Sum and product of the roots of 3x^2 + 9x

Step by step

The question

Question: find the sum and the product of the roots of 3x² + 9x − 2 = 0.

3x² + 9x − 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 = 3, b = 9, c = −2.

= a = 3, b = 9, 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 9, over 3.

That is −3.

r + s = −93 = −3

Careful!

The sum has a minus: minus b over a.

9/3 has the wrong sign.

Careful: not r + s = 93 ✗

3 Product of the roots

The product is c/a: −2/3.

rs = −23

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

r + s = −3, rs = −23

4 Check

Let's check with the roots from the formula.

the formula: x = −9 + √1056 and −9 − √1056

Added, plus root 105 and minus root 105 cancel.

−9 plus −9, over 6, is −3.

sum: −9 + −96 = −3

Multiplied, the top is −9 squared minus 105: -24.

Over 36 that is −2/3.

The same!

So the answer is right.

product: 81 − 10536 = −23

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer