Sum and product of the roots of 5x^2 + 2x

Step by step

The question

Question: find the sum and the product of the roots of 5x² + 2x − 1 = 0.

5x² + 2x − 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 = 5, b = 2, c = −1.

= a = 5, b = 2, 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 2, over 5.

That is −2/5.

r + s = −25 = −25

Careful!

The sum has a minus: minus b over a.

2/5 has the wrong sign.

Careful: not r + s = 25 ✗

3 Product of the roots

The product is c/a: −1/5.

rs = −15

So the sum of the roots is −2/5 and the product is −1/5.

r + s = −25, rs = −15

4 Check

Let's check with the roots from the formula.

the formula: x = −2 + √2410 and −2 − √2410

Added, plus root 24 and minus root 24 cancel.

−2 plus −2, over 10, is −2/5.

sum: −2 + −210 = −25

Multiplied, the top is −2 squared minus 24: -20.

Over 100 that is −1/5.

The same!

So the answer is right.

product: 4 − 24100 = −15

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer