91Ó°ÊÓ

In quadratic equations, the discriminant is a key component that helps us determine the nature of the roots. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated as \( b^2 - 4ac \). It provides essential insights into the type of roots we can expect the equation to have.

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, and this root is repeated. This is also known as a double root.
• If \( D < 0 \), the equation has two complex roots.

In the given exercise, our goal was to find a value of \( k \) such that the equation would have exactly one real root, meaning the discriminant must equal zero. By equating \( b^2 - 4ac = 0 \), we isolate the variable \( k \) to find that exact condition.

A real root is a solution to a quadratic equation that is a real number. Real roots can be seen visually as the points where the graph of the quadratic function intersects the x-axis. Quadratic equations can have zero, one, or two real roots based on the value of the discriminant.

• If there are no intersections, the quadratic function has no real roots (complex roots).
• One point of intersection indicates exactly one real root, which occurs when the discriminant is zero.
• Two intersections correspond to two distinct real roots.

In our case, the problem specifies that the equation should have one real root. This is an important scenario often characterized by a parabolic graph that just touches the x-axis at its vertex. By solving the equation \( 2k - 72 = 0 \), we confirm that \( k = 36 \) ensures the equation has this single intersection point, making \( k = 36 \) the required value for the equation to have exactly one real root.

The quadratic formula is a universal method for solving quadratic equations. When we have a quadratic equation \( ax^2 + bx + c = 0 \), the roots of the equation (i.e., the values of \( x \) that satisfy the equation) can be found using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula incorporates the discriminant, \( b^2 - 4ac \), directly into the calculation of the roots. Depending on the value of the discriminant:

• If \( D = 0 \), the formula shows that both roots are the same, hence there is exactly one real root.
• If \( D > 0 \), the formula yields two distinct real roots.
• If \( D < 0 \), the value under the square root becomes negative, leading to complex roots.

In our exercise, even though we caried out different calculations to focus on setting the discriminant to zero, the quadratic formula could still directly verify the condition of having one real root once \( k \) was determined.