Math Problem Statement

Consider f(x) = 2kx^2 - 4kx + 1 = 0, for k ≠ 0. The equation f(x) = 0 has two equal roots. (a) Find the value of k. (b) The line y = p intersects the graph of f. Find all possible values of p.

Solution

Let's solve the problem step by step.

Given:

f(x)=2kx24kx+1f(x) = 2kx^2 - 4kx + 1 with k0k \neq 0. We know that f(x)=0f(x) = 0 has two equal roots.

Part (a): Find the value of kk.

To find when a quadratic equation has two equal roots, we use the discriminant. For a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the discriminant (Δ\Delta) is:

Δ=b24ac\Delta = b^2 - 4ac

If the equation has two equal roots, the discriminant must be zero:

Δ=0\Delta = 0

For the function f(x)=2kx24kx+1f(x) = 2kx^2 - 4kx + 1:

  • a=2ka = 2k
  • b=4kb = -4k
  • c=1c = 1

So, the discriminant becomes:

Δ=(4k)24(2k)(1)=16k28k\Delta = (-4k)^2 - 4(2k)(1) = 16k^2 - 8k

Setting the discriminant to zero:

16k28k=016k^2 - 8k = 0

Factor out 8k8k:

8k(2k1)=08k(2k - 1) = 0

Since k0k \neq 0, we have:

2k1=02k - 1 = 0

k=12k = \frac{1}{2}

Answer for Part (a):

k=12k = \frac{1}{2}

Part (b): Find all possible values of pp.

We are asked where the line y=py = p intersects the graph of f(x)f(x). To find this, set f(x)=pf(x) = p:

2kx24kx+1=p2kx^2 - 4kx + 1 = p

Substituting k=12k = \frac{1}{2}:

212x2412x+1=p2 \cdot \frac{1}{2}x^2 - 4 \cdot \frac{1}{2}x + 1 = p

x22x+1=px^2 - 2x + 1 = p

(x1)2=p(x - 1)^2 = p

Since (x1)2(x - 1)^2 is always non-negative, pp must be non-negative as well. Thus:

p0p \geq 0

Answer for Part (b):

p0p \geq 0

Would you like further details or have any questions?

5 Related Questions:

  1. How do you determine if a quadratic equation has no real roots using the discriminant?
  2. What happens if k=0k = 0 in the given equation?
  3. Can pp be a negative value in any case for this function?
  4. How can we confirm if the roots of a quadratic equation are real and distinct?
  5. What is the geometric interpretation of a line intersecting a parabola?

Tip:

Remember that the discriminant determines the nature of the roots: zero indicates a double root, positive indicates two distinct roots, and negative indicates no real roots.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Discriminant

Formulas

Discriminant: Δ = b^2 - 4ac
Quadratic equation: ax^2 + bx + c = 0

Theorems

Properties of Discriminants

Suitable Grade Level

High School (Grades 10-12)