What is the value of c?

Question

The full question:

The function ƒ is defined by ƒ(x) = 2x² + 5x - c, where c is a constant. In the xy-plane, the graph of ƒ intersects the x-axis at the points (1/2, 0) and (-3,0). What is the value of c?

 

I got confused with this. Please help. Thank you!

Answer 1

Forget about intercepts, factoring and working backwards;  the problem gives two points on the curve.  They are:  (1/2,0) and (-3.0).

That means [remembering that y=f(x)]:
          0 = 2(1/2)^2 + 5(1/2) - c         and           0 = 2(-3)^2 + 5(-3) – c
          0 = (1/2)  + 5/2  - c                 and           0 = 18 – 15 – c
          c = 3                                       and           c = 3

Actually, any one point on the curve would have been enough to determine the value of c, since it is a CONSTANT.

The function is:    f(x) = 2x^2 + 5x -3

Answer 2

We know that when we set f(x) equal to zero, and solve for x we get our x-intercepts.  Since we know our x-intercepts, we can work backwards.

 

 

Set f(x) equal to zero and write the given function as a product of two binomials using the roots.

 

0 = (2x + q)(x + 3)             knowing that multiplying the first terms gives us 2x².

 

 

Set these factors equal to zero:              (2x + q)            and          (x + 3)

 

 

If     x + 3 = 0   , then the value of x is -3, one of our x-intercepts.    We need to solve for q from the other factor.

 

 

2x + q = 0

 

 

Let's plug x=1/2 for this one and solve for q.

 

2(1/2) + q = 0

 

1 + q = 0

 

       q = -1

 

 

Plug in this value of q into the factor.  Multiply the factor we used to solve for q by (x + 3).

 

f(x) = (2x - 1)(x + 3)

 

f(x) = 2x² + 6x - x - 3

 

f(x) = 2x² + 5x - 3

 

 

The value of c is 3.