Sum and product of the roots of 4x^2 - 8x + 3 = 0 Step by Step

Step by step

The question

Question: find the sum and the product of the roots of 4x² − 8x + 3 = 0.

4x² − 8x + 3 = 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 = 4, b = −8, c = 3.

= a = 4, b = −8, c = 3

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 (−8), over 4.

That is 2.

r + s = −(−8)4 = 2

Careful!

The sum has a minus: minus b over a.

−8/4 has the wrong sign.

Careful: not r + s = −84 ✗

3 Product of the roots

The product is c/a: 3/4.

rs = 34

So the sum of the roots is 2 and the product is 3/4.

r + s = 2, rs = 34

4 Check

Let's check.

Solving the equation gives x = 3/2 and x = 1/2.

the roots: x = 32 and x = 12

Their sum is 2 and their product is 3/4.

The same!

So the answer is right.

sum: 32 + 12 = 2 product: 32 × 12 = 34

Let's look back.

First: read a, b and c.

Then: sum of the roots.

Then: product of the roots.

Last: check.

Answer