Sum and product of the roots of x^2 - 4x + 1 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of x² − 4x + 1 = 0.

x² − 4x + 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 = 1, b = −4, c = 1.

= a = 1, b = −4, 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 (−4), over 1.

That is 4.

r + s = −(−4) = 4

Careful!

The sum has a minus: minus b over a.

−4/1 has the wrong sign.

Careful: not r + s = −41 ✗

3 Product of the roots

The product is c/a: 1/1.

That is 1.

rs = 1

So the sum of the roots is 4 and the product is 1.

r + s = 4, rs = 1

4 Check

Let's check with the roots from the formula.

the formula: x = 4 + √122 and 4 − √122

Added, plus root 12 and minus root 12 cancel.

4 plus 4, over 2, is 4.

sum: 4 + 42 = 4

Multiplied, the top is 4 squared minus 12: 4.

Over 4 that is 1.

The same!

So the answer is right.

product: 16 − 124 = 1

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer