a If p and m are two roots of a quadratic equation, then (x-p) and (x-m) are the factors of the equation.
B
b Multiply the factors (x-p) and (x-m). What can you say about the coefficients and the constant?
C
c Consider steps you took in the previous parts.
A
aTable:
Roots
Factors
2, 5
(x-2), (x-5)
1, 9
(x-1), (x-9)
- 1, 3
(x+1), (x-3)
0, 6
x, (x-6)
1/2, 7
(x-1/2), (x-7)
- 2/3, 4
(x+2/3), (x-4)
B
bTable:
Roots
Factors
Equation
2, 5
(x-2), (x-5)
x^2-7x+10=0
1, 9
(x-1), (x-9)
x^2-10x+9=0
- 1, 3
(x+1), (x-3)
x^2-2x-3=0
0, 6
x, (x-6)
x^2-6x=0
1/2, 7
(x-1/2), (x-7)
2x^2-15x+ 7=0
- 2/3, 4
(x+2/3), (x-4)
3x^2-10x-8=0
C
c
Equation: x^3-6x^2+11x-6=0
No, see solution.
Practice makes perfect
a We will use the following fact.
If pand mare two roots of a quadratic
equation, then(x- p)and(x- m)
are the factors of the equation.
Now, let's complete the second column of the table.
Roots
Factors
2, 5
(x- 2), (x- 5)
1, 9
(x- 1), (x- 9)
- 1, 3
(x-( - 1)), (x- 3) or (x+1), (x-3)
0, 6
(x- 0), (x- 6) or x, (x-6)
1/2, 7
(x- 1/2), (x- 7)
- 2/3, 4
(x-( - 2/3)), (x- 4) [0.5em] or [0.5em] (x+2/3), (x-4)
b Knowing the factors of a quadratic equation, we can write the quadratic equation by multiplying them.
(x- p)(x- m)=0
⇕
x^2-(p+m)x+pm=0
As we can see, the coefficient of x is the sum of the roots and the constant term is their product. With this in mind, let's complete the last column.
Roots
Factors
Equation
2, 5
(x-2), (x-5)
(x-2)(x-5)=0 x^2-7x+10=0
1, 9
(x-1), (x-9)
(x-1)(x-9)=0 x^2-10x+9=0
- 1, 3
(x+1), (x-3)
(x+1)(x-3)=0 x^2-2x-3=0
0, 6
x, (x-6)
x(x-6)=0 x^2-6x=0
1/2, 7
(x-1/2), (x-7)
(x-1/2)(x-7)=0 x^2-15/2x+ 7/2=0
- 2/3, 4
(x+2/3), (x-4)
(x+2/3)(x-4)=0 x^2-10/3x- 8/3=0
To get rid of the fractions in the last two equations, we can multiply both sides of the equations by the denominator of the fractions.
Roots
Factors
Equation
2, 5
(x-2), (x-5)
x^2-7x+10=0
1, 9
(x-1), (x-9)
x^2-10x+9=0
- 1, 3
(x+1), (x-3)
x^2-2x-3=0
0, 6
x, (x-6)
x^2-6x=0
1/2, 7
(x-1/2), (x-7)
2x^2-15x+ 7=0
- 2/3, 4
(x+2/3), (x-4)
3x^2-10x-8=0
c To write an equation with three roots, we first determine the factors by using the roots. Then, we multiply these factors and set it equal to zero, similar to what we did for the equation with two roots. Let's write an equation with roots 1, 2, and 3.
Factors: [-0.8em]
(x- 1), (x- 2), (x-3)
We will now multiply these factors and equate it to zero.
