Sum and product of the roots of 3x^2 - 7x + 2 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of 3x² − 7x + 2 = 0.

3x² − 7x + 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 = −7, c = 2.

= a = 3, b = −7, 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 (−7), over 3.

That is 7/3.

r + s = −(−7)3 = 73

Careful!

The sum has a minus: minus b over a.

−7/3 has the wrong sign.

Careful: not r + s = −73 ✗

3 Product of the roots

The product is c/a: 2/3.

rs = 23

So the sum of the roots is 7/3 and the product is 2/3.

r + s = 73, rs = 23

4 Check

Let's check.

Solving the equation gives x = 2 and x = 1/3.

the roots: x = 2 and x = 13

Their sum is 7/3 and their product is 2/3.

The same!

So the answer is right.

sum: 2 + 13 = 73 product: 2 × 13 = 23

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer