discriminant graphs
The Discriminant of a Quadratic Equation
Part 5

Today I’d like to solve the problem about discriminants from the last post. You can review parts 1, 2, 3, and 4 of our discussion on the discriminant here:

The Discriminant of a Quadratic Equation – Part 1
The Discriminant of a Quadratic Equation – Part 2
The Discriminant of a Quadratic Equation – Part 3
The Discriminant of a Quadratic Equation – Part 4

Last time, I asked you to solve the following problem:

Example: You are given the following system of equations.

dx + ey = f
y = x2

where d, e, and f are integers. For which of the following will there be more than one (xy) solution, with real-number coordinates for the system?

Ae2 + 4df < 0
B. e2 – 4df < 0
C. d2 – 4ef < 0
D. e2 – 4df > 0
E. d2 + 4ef > 0

Make sure to try this problem yourself before reading the following solution.

Solution: By the second equation, we have y = x2, so we can replace y by xin the first equation to get

dx + ex2 = f

Writing this quadratic equation in general form gives the following.

 ex2 + dx – f = 0

So, we have aebd, and c f . Thus, the discriminant is

Δ = b2 – 4ac = d2 – 4e( f) = d2 + 4ef

We want the quadratic equation to have two real solutions. Therefore, the discriminant must be positive. So, we must have

d2 + 4ef > 0

This is choice E

 

If you liked this article, please share it with your Facebook friends:

Quadratic Equations Facebook Share Button

Get 800 SAT Math Prep Facebook Link Get 800 SAT Math Prep Twitter Link Get 800 SAT Math Prep YouTube Link Get 800 SAT Math Prep LinkedIn Link Get 800 SAT Math Prep Pinterest Link

Comments

comments