Understanding Quadratic Equations and Real Roots

The problem asks us to find the greatest value of \(k\) for the quadratic equation \(2x^2 - 4x + k = 0\) such that it has real roots. A quadratic equation is generally written in the form \(ax^2 + bx + c = 0\). For an equation to have real roots, its discriminant must be non-negative (greater than or equal to zero).

Determining the Discriminant Condition

The discriminant, denoted by \(\Delta\), is calculated using the formula:
\( \Delta = b^2 - 4ac \)

For a quadratic equation to have real roots, the condition is:
\( \Delta \ge 0 \)

If \(\Delta > 0\), the equation has two distinct real roots.

If \(\Delta = 0\), the equation has exactly one real root (or two equal real roots).

If \(\Delta < 0\), the equation has no real roots (it has two complex conjugate roots).

Applying Discriminant to \(2x^2 - 4x + k = 0\)

In the given equation, \(2x^2 - 4x + k = 0\), we can identify the coefficients:

• \(a = 2\)
• \(b = -4\)
• \(c = k\)

Now, let's calculate the discriminant for this specific equation:

\( \Delta = (-4)^2 - 4(2)(k) \)

\( \Delta = 16 - 8k \)

Solving for the Greatest Value of \(k\)

To ensure the equation has real roots, we must satisfy the condition \(\Delta \ge 0\). Substitute the calculated discriminant:

\( 16 - 8k \ge 0 \)

Now, we need to solve this inequality for \(k\).

1. Add \(8k\) to both sides:
   \( 16 \ge 8k \)
2. Divide both sides by 8 (since 8 is positive, the inequality sign remains the same):
   \( \frac{16}{8} \ge k \)
3. Simplify the fraction:
   \( 2 \ge k \)

This inequality, \(2 \ge k\), means that \(k\) must be less than or equal to 2 (\(k \le 2\)).

The question asks for the greatest value of \(k\) that satisfies this condition. Since \(k\) must be less than or equal to 2, the greatest possible value for \(k\) is 2.