Worked examples
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.
▶ Watch it solved
The question
Question: find the sum and the product of the roots of 2x² − 6x + 1 = 0.
2x² − 6x + 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 = 2, b = −6, c = 1.
= a = 2, b = −6, 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 (−6), over 2.
That is 3.
r + s = −(−6)2 = 3
Careful!
The sum has a minus: minus b over a.
−6/2 has the wrong sign.
Careful: not r + s = −62 ✗
3 Product of the roots
The product is c/a: 1/2.
rs = 12
So the sum of the roots is 3 and the product is 1/2.
r + s = 3, rs = 12
4 Check
Let's check with the roots from the formula.
the formula: x = 6 + √284 and 6 − √284
Added, plus root 28 and minus root 28 cancel.
6 plus 6, over 4, is 3.
sum: 6 + 64 = 3
Multiplied, the top is 6 squared minus 28: 8.
Over 16 that is 1/2.
The same!
So the answer is right.
product: 36 − 2816 = 12
Let's look back.
First: read a, b and c.
Then: sum of the roots.
Then: product of the roots.
Last: check.
Answer r + s = 3, rs = 12
▶ Watch it solved
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 r + s = 2, rs = 34
▶ Watch it solved
