91Ó°ÊÓ

The discriminant is a crucial part of quadratic equations. It helps determine the nature and number of roots of the equation. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated with the formula: \[ D = b^2 - 4ac \] The value of \( D \) tells us how many real solutions the equation has:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there's exactly one real solution, called a repeated or double root.
• If \( D < 0 \), no real solutions exist, implying complex solutions.

In our initial problem, the calculation of the discriminant was \( D = (-20)^2 - 4(4)(25) = 0 \). This result immediately informs us that our equation will have a single real solution.

Real solutions of a quadratic equation are the x-values that satisfy the equation. After determining the discriminant, the real solutions can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] With \( D = 0 \), the formula simplifies as the \( \pm \) component vanishes, giving just one solution. For our equation \( 4x^2 - 20x + 25 = 0 \), we substituted \( b = -20 \), \( D = 0 \), and \( a = 4 \) into the formula, resulting in \( x = \frac{20}{8} = 2.5 \). Therefore, the quadratic equation has one real solution at \( x = 2.5 \). Understanding real solutions helps students grasp the concept of roots in quadratic equations, particularly in how they relate to the equation's graph.

Graphically interpreting a quadratic equation involves graphing the related function and observing its intersection with the x-axis. The equation \( ax^2 + bx + c = 0 \) forms a parabola when graphed. How the parabola interacts with the x-axis tells us about the solutions.

• If the parabola intersects the x-axis at two points, the equation has two real solutions.
• If it touches the x-axis at a single point, there is one real solution, a situation known as a "repeated root" or "vertex touching the axis" phenomenon.
• If it does not intersect the x-axis, the equation has no real solutions but complex ones instead.

In this case, our parabola represented by \( 4x^2 - 20x + 25 \) touches the x-axis at \( x = 2.5 \). This touching point signifies that the equation has a repeated root at \( x = 2.5 \), coinciding with our calculated solution. Graphs can be an intuitive way to verify solutions and enhance understanding.