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The quadratic formula is a powerful tool that helps us find the roots or zeros of quadratic equations. A quadratic equation has the form \( ax^2 + bx + c = 0 \). The solution, using the quadratic formula, is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The formula allows us to calculate the exact values where the quadratic function crosses the x-axis, also known as the zeros. Understanding this formula is crucial to solving quadratic equations effectively.

• The symbol \( \pm \) means that there will be two solutions, one for addition and one for subtraction.
• The part under the square root, \( b^2 - 4ac \), is called the discriminant.
• The discriminant helps us determine the nature of the roots (e.g., real or complex).

Once you identify the coefficients from the quadratic equation, simply plug them into the formula and solve for \( x \). The step is straightforward but requires correct arithmetic operations.

The zeros of a function are the values of \( x \) where the function equals zero. For quadratic functions, these are the points where the parabola intersects the x-axis. Finding zeros is essential in various fields, like physics or engineering, since it gives insight into significant behaviors of the system modeled by the function.To find zeros, as seen in the original problem, we use the quadratic formula. The two values \( x_1 = \frac{5+\sqrt{37}}{6} \) and \( x_2 = \frac{5-\sqrt{37}}{6} \) are the zeros. These values tell us exactly where the function switches its direction on the x-axis.

• If the discriminant \( b^2 - 4ac > 0 \), the function has two distinct real zeros.
• If \( b^2 - 4ac = 0 \), the function has exactly one zero or a repeated root.
• If \( b^2 - 4ac < 0 \), the function has no real zero, but two complex zeros.

The calculation of zeros plays a vital role in sketching the curve of the function and understanding its geometry.

Every quadratic equation consists of coefficients that define its specific shape and position on the graph. These are the values \( a \), \( b \), and \( c \) in the equation \( ax^2 + bx + c = 0 \).

• \( a \): the coefficient in front of \( x^2 \) determines the width and direction of the parabola (opened upward if \( a > 0 \), downward if \( a < 0 \)).
• \( b \): the coefficient of \( x \) affects the parabola's placement along the x-axis and its symmetry.
• \( c \): the constant term, which gives the y-intercept of the quadratic graph.

Identifying these coefficients is the first step in applying the quadratic formula. It is crucial to get these correct so that the zeros calculated will give the true intersection points of the function with the x-axis. In our example equation, \( a = 3 \), \( b = -5 \), and \( c = -1 \). Properly recognizing them sets the foundation for accurately solving quadratic equations.