Sum and product of the roots of 2x^2 - 6x + 1 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of 2x² − 6x + 1 = 0.

2x² − 6x + 1 = 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 = −6, c = 1.

= a = 2, b = −6, c = 1

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 (−6), over 2.

That is 3.

r + s = −(−6)2 = 3

Careful!

The sum has a minus: minus b over a.

−6/2 has the wrong sign.

Careful: not r + s = −62 ✗

3 Product of the roots

The product is c/a: 1/2.

rs = 12

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

r + s = 3, rs = 12

4 Check

Let's check with the roots from the formula.

the formula: x = 6 + √284 and 6 − √284

Added, plus root 28 and minus root 28 cancel.

6 plus 6, over 4, is 3.

sum: 6 + 64 = 3

Multiplied, the top is 6 squared minus 28: 8.

Over 16 that is 1/2.

The same!

So the answer is right.

product: 36 − 2816 = 12

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer