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The quadratic formula is a powerful tool that helps us solve quadratic equations, which are polynomials of degree two. These equations commonly look like this:

• \[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown quantity we need to find.
To find the solutions, or "roots," we use the quadratic formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The formula might seem complex, but with practice, it becomes a straightforward method!

Steps for Using the Quadratic Formula

• First, identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant, which is the portion under the square root: \( b^2 - 4ac \).
• Substitute the values into the quadratic formula and solve for \( x \).

The discriminant is a key component in the quadratic formula and has an essential role in determining the nature of the roots. It is given by the expression:

• \[ b^2 - 4ac \]

where \( b \), \( a \), and \( c \) are coefficients from the quadratic equation.

Understanding the Discriminant

The value of the discriminant provides crucial information:

• If it is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, known as a repeated or double root.
• If it is negative, the equation has no real roots, but two complex roots.

In the solution of \( 25f^2 - 80f + 15 = 0 \), the discriminant was found to be 4900, indicating two real roots for the equation.

Finding the roots of a quadratic equation means solving for the values of \( x \) that satisfy the equation. These roots are the solutions where the equation equals zero.For example, in our equation

• \[ 25f^2 - 80f + 15 = 0 \]

we can find the roots by using the quadratic formula. The process involves calculating two potential answers based on adding and subtracting the square root of the discriminant.

Calculating the Roots

In our example:

• First solution: Substitute and solve \( \frac{80 + 70}{50} = 3 \)
• Second solution: Substitute and solve \( \frac{80 - 70}{50} = 0.2 \)

Thus, the quadratic equation has two roots: \( f = 3 \) and \( f = 0.2 \). These solutions are points where the function \( 25f^2 - 80f + 15 \) crosses the axis.