Sum and product of the roots of x^2 + 7x + 10 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of x² + 7x + 10 = 0.

x² + 7x + 10 = 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 = 7, c = 10.

= a = 1, b = 7, c = 10

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 1.

That is −7.

r + s = −7 = −7

Careful!

The sum has a minus: minus b over a.

7/1 has the wrong sign.

Careful: not r + s = 71 ✗

3 Product of the roots

The product is c/a: 10/1.

That is 10.

rs = 10

So the sum of the roots is −7 and the product is 10.

r + s = −7, rs = 10

4 Check

Let's check.

Solving the equation gives x = −2 and x = −5.

the roots: x = −2 and x = −5

Their sum is −7 and their product is 10.

The same!

So the answer is right.

sum: −2 + (−5) = −7 product: −2 × (−5) = 10

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer