Determine a Quadratic Equation Given its roots

Question

Determine the equation in standard form for each quadratic function.

 

a) x-intercepts -2+√5 and -2 -√5, containing the point (-4,5)

 

 

 

 

Answer 1

Find two factors that contain the x-intercepts and multiply them.  Then set that product equal to f(x)

 

f(x) = C (x + [2 - √5])(x + [2 + √5])

 

 

Next, we expand the function using FOIL.  The brackets will guide you through the process.

 

f(x) = C[x² + x(2 + √5) + x(2 - √5) + (4 - 5)]

 

f(x) = C(x² + 4x - 1)

 

 

Now we find the value of C using the condition    f(-4)=5.

 

5 = C((-4)² + 4(-4) - 1)

 

5 = C(16 - 16 - 1)

 

5 = -33C

 

C = -5/33

 

 

Now just plug this value of C into the function.

 

f(x) = (-5/33)(x² + 4x - 1)

 

 

Then to get it in standard form, distribute the value of C to all the terms in parenthesis.