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The quadratic formula is a powerful tool used to find the roots of any quadratic equation, which takes the general form: \( ax^2 + bx + c = 0 \). The formula itself is defined as: \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
This means that once you know the coefficients \( A \), \( B \), and \( C \), you can determine the solutions for \( x \). The symbol \( \pm \) indicates you will typically have two solutions: one involving addition and the other subtraction under the square root sign.
In the given exercise, we identify that \( A = 4 \), \( B = -20 \), and \( C = 25 \). These coefficients will allow us to compute the discriminant and subsequently find the roots of the quadratic.

The discriminant is a component of the quadratic formula represented by \( B^2 - 4AC \). It is crucial because it helps us determine the nature of the roots without solving for them directly.
Here's how the discriminant works:

• If it is positive, there are two distinct real roots.
• If it is zero, there is one real root, called a double root.
• If it is negative, the roots are complex numbers.

In our exercise, we calculated the discriminant as follows: \( (-20)^2 - 4 \times 4 \times 25 = 400 - 400 = 0 \). This indicates that this quadratic has a double root, which simplifies the factorization process.

A double root occurs when both solutions from the quadratic formula are identical. This happens when the discriminant equals zero. A double root implies that the graph of the quadratic touches the x-axis at only one point.
For our quadratic equation, we used the formula: \( x = \frac{-B}{2A} \) since \( \sqrt{0} \) becomes zero, and the ± part can be ignored.
With \( B = -20 \) and \( A = 4 \), we find the double root to be \( x = \frac{20}{8} = 2.5 \). Understanding the double root allows us to express the quadratic in a simplified factored form with one repeating factor.

In this context, factored form refers to expressing the quadratic as a product of its linear factors. When a quadratic has a double root \( r \), it can be represented as \((x - r)^2\).
For our equation with double root \( 2.5 \), the initial expression we consider is \((x - 2.5)^2\). However, our original quadratic \(4x^2 - 20x + 25\) has a leading coefficient other than 1, which means we need to adjust for this.
By factoring \((x - 2.5)(x - 2.5) = (2x - 5)^2\), it ensures each term corresponds correctly when expanded. Expanding confirms this: \((2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25\).
Thus, verifying the factored form is correct and aligns perfectly with the original quadratic expression.