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The quadratic formula is a crucial tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula is used to find the values of \( x \) that satisfy the equation, known as the roots. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Understanding this formula is important because it helps us determine the number and type of solutions (roots) that a quadratic equation may have. The formula requires three coefficients from the equation:

• \( a \), the coefficient of \( x^2 \)
• \( b \), the coefficient of \( x \)
• \( c \), the constant term

These coefficients are substituted into the formula to calculate the solutions for \( x \). Making sure you know how to identify \( a \), \( b \), and \( c \) from a quadratic equation is essential for using the quadratic formula effectively.

The discriminant is a part of the quadratic formula that helps us determine the nature of the roots of a quadratic equation. It is represented by \( \Delta \) (Delta) and calculated as: \[ \Delta = b^2 - 4ac \] This value tells us about the type of roots we can expect:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one repeated real root (also known as a double root).
• If \( \Delta < 0 \), there are no real roots, but two complex roots instead.

In our example, the discriminant turned out to be zero, which means the quadratic equation has a repeated real root. Recognizing how the discriminant impacts the number of roots is a key skill when solving quadratic equations.

The roots of a polynomial are the solutions to the equation derived from setting the polynomial equal to zero. For quadratic polynomials, like \( ax^2 + bx + c \), the roots can typically be found using the quadratic formula. The roots indicate where the graph of the polynomial intersects the x-axis. In solving \( f(x) = 16x^2 - 40x + 25 \), we're looking for the \( x \) values where the function equals zero. After calculating using the quadratic formula, we find the specific values that show where the graph touches or crosses the x-axis:

• When two distinct roots are found, the graph cuts the x-axis at two points.
• A single repeated root means the graph just touches the x-axis at one point.

These points are critical in understanding the visual representation of quadratic functions.

A repeated root, also known as a double root, occurs when a quadratic equation has a discriminant of zero. This suggests the equation has only one solution where the graph of the polynomial just "touches" the x-axis but does not cross it. For the quadratic equation \( f(x) = 16x^2 - 40x + 25 \), the repeated root was found to be \( x = \frac{5}{4} \). This means both roots are the same, and the parabola represented by the quadratic equation is tangent to the x-axis at this point. Understanding repeated roots is useful in graphing and interpreting quadratic functions because it describes a special case where the graph does not intersect the x-axis twice as it usually might with distinct roots.