Sum and product of the roots of x^2 - 5x + 6 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of x² − 5x + 6 = 0.

x² − 5x + 6 = 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 = −5, c = 6.

= a = 1, b = −5, c = 6

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 (−5), over 1.

That is 5.

r + s = −(−5) = 5

Careful!

The sum has a minus: minus b over a.

−5/1 has the wrong sign.

Careful: not r + s = −51 ✗

3 Product of the roots

The product is c/a: 6/1.

That is 6.

rs = 6

So the sum of the roots is 5 and the product is 6.

r + s = 5, rs = 6

4 Check

Let's check.

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

the roots: x = 3 and x = 2

Their sum is 5 and their product is 6.

The same!

So the answer is right.

sum: 3 + 2 = 5 product: 3 × 2 = 6

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer