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The quadratic formula is a vital tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). To apply the quadratic formula, you use:

• \( y = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

This formula allows you to find the roots or solutions of the quadratic equation. Here, the letters \( a \), \( b \), and \( c \) are the coefficients of the equation that must be known. The "+" and "-" signs indicate that there are two solutions—one using addition and the other using subtraction.

This method is especially helpful because it works for all types of quadratic equations, whether they can be factored or not. It's a universal solution method. Remember, the value you calculate using the quadratic formula will give you the points where the equation graphically crosses the x-axis in a parabolic graph.

The standard form of a quadratic equation is represented as \( ax^2 + bx + c = 0 \). Writing equations in this form is essential before applying the quadratic formula. Let's break it down:

• \( a \): It is the coefficient of the squared term \( x^2 \) or \( y^2 \) in this context. It dictates the parabola's "width" and direction (narrow or wide, up or down).
• \( b \): This is the coefficient of the linear term \( x \) or \( y \). It affects the location of the parabola, especially in relation to its symmetry.
• \( c \): This is the constant term. It provides the y-intercept of the parabola.

Before you start solving, ensure that your quadratic equation is rearranged into this form. For instance, from the original equation \( y^2 - 0.5y = -0.06 \), we adjust it to \( y^2 - 0.5y + 0.06 = 0 \) by moving all terms to one side, making it a zero equation.

The discriminant of a quadratic equation is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots:

• If the discriminant \( (b^2 - 4ac) \) is positive, the equation has two real and distinct solutions.
• If the discriminant is zero, there is exactly one real solution. This occurs because the vertex of the parabola touches the x-axis.
• If the discriminant is negative, the solutions are complex or imaginary, as the graph doesn't cross the x-axis.

In the given exercise, the calculated discriminant is \( 0.01 \) (since \( b^2 = 0.25 \) and \( 4ac = 0.24 \)), meaning the equation has two real solutions. Calculating the discriminant is crucial as it immediately tells you the potential type and number of solutions you can expect before crunching all the numbers in the quadratic formula.