91Ó°ÊÓ

When dealing with a quadratic equation, the quadratic formula is a powerful tool that can find the solutions easily. The formula is:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
This might look complex at first. However, breaking it down step-by-step makes it manageable.
First, identify the coefficients from your equation \(ax^2 + bx + c = 0\). For the equation given, 25x^2 - 20x + 4 = 0, we have:

• a = 25
• b = -20
• c = 4

Once you have these values, you can substitute them into the quadratic formula. Pay close attention to the signs, especially with negative numbers. This ensures you get the correct solutions.

The discriminant, denoted as \Delta (Delta)\, is a crucial component in determining the nature of the solutions for a quadratic equation. It is found inside the square root of the quadratic formula:

• Discriminant formula: \( \Delta = b^2 - 4ac \)

In our specific problem, using a = 25, b = -20, and c = 4, we calculate:
\[ \Delta = (-20)^2 - 4(25)(4) \] \[ \Delta = 400 - 400 \] \[ \Delta = 0 \]
The value of the discriminant tells us the nature of the solutions:

• \( \Delta > 0 \) : Two distinct real solutions
• \( \Delta = 0 \) : One real solution
• \( \Delta < 0 \) : No real solutions (complex or imaginary solutions)

In this exercise, \Delta is 0, indicating that we have exactly one real solution.

With a discriminant of zero, we have precisely one real solution for the quadratic equation. The steps to find this solution are conscientious and should be followed attentively.
Substitute the values of a, b, and \Delta into the quadratic formula again:
\[ x = \frac{-b \pm \sqrt{0}}{2a} \] \[ x = \frac{20 \pm 0}{50} \] \[ x = \frac{20}{50} \] \[ x = \frac{2}{5} \]
Because \sqrt{0} equals zero, there's only one value for x, which simplifies the equation enormously. Thus, x = \frac{2}{5} is the sole real solution.
This means the quadratic equation 25x^2 - 20x + 4 = 0 has one real root, specifically at x = \frac{2}{5}. Understanding how to apply the quadratic formula and interpret the discriminant will help you solve similar quadratic equations confidently.