quadratic equation

Question

Write the quadratic equation whose roots are
2
and
−3
, and whose leading coefficient is
5
.
(Use the letter
x
to represent the variable.)

Answer 1

Hi Maryalice:

 

Arturo's answer and method are simpler

but, if you wish to look at this from another

angle you can set up your equation as:

 

f(x) = 5x² + bx + c

With roots at 2 and -3 we have 2 simultaneous equations:

 

5(2)² + 2b + c = 0

20 + 2b + c = 0

2b + c = -20     {equation 1}

 

5(-3)² - 3b + c = 0

45 - 3b + c = 0

3b - c = 45      {equation 2}

 

2b + c = -20

3b - c = 45

------------

5b = 25

b = 5

 

2(5) +c = -20

c = -30

 

f(x) = 5x² + 5x - 30

 

This is the same as Arturo's answer:

5x² + 5x - 30 = 5(x² + x - 6) = 5(x-2)(x+3)

 

Answer 2

f(x) = a(x - x₁)(x - x₂)

 

x₁ = 2

x₂ = -3

a = 5

 

f(x) = 5(x - 2)[x - (-3)]

 

f(x) = 5(x - 2)(x + 3)