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Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \). These equations are used to model various real-life scenarios, ranging from calculating areas to predicting outcomes in physics and economics. Quadratics are characterized by the highest exponent of the variable \( x \) being 2, and this distinct form results in a parabolic graph when plotted on a coordinate plane.
Understanding the components:

• \( a \) - the coefficient of \( x^2 \), which affects the "opening" direction and width of the parabola.
• \( b \) - the coefficient of \( x \), which influences the parabola's position along the x-axis.
• \( c \) - the constant term, which dictates the point where the parabola intersects the y-axis.

To solve these equations, we often use methods like factoring, completing the square, or using the quadratic formula. The quadratic formula becomes indispensable when the equation does not factor easily.

The discriminant is a vital part of solving quadratic equations as it helps us understand the nature of the solutions. For any quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by the expression \( b^2 - 4ac \). This value determines how many and what type of solutions the quadratic equation has.
Let's break down its significance:

• A positive discriminant (\( D > 0 \)) indicates two distinct real number solutions.
• A zero discriminant (\( D = 0 \)) reveals there is exactly one real solution, also called a repeated or double root.
• A negative discriminant (\( D < 0 \)) shows there are no real solutions, but instead two complex solutions.

In this exercise, calculating the discriminant helps verify that the quadratic equation has real solutions. Here, \( D = \frac{7}{9} \), a positive value, suggesting two real and distinct solutions. It's a quick check before performing further calculations.

Real number solutions for quadratic equations are the roots where the graph of a quadratic equation intersects the x-axis. These solutions can be found using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides the tool for calculating these roots when the quadratic equation resists simpler solving methods like factoring.
When using the quadratic formula:

• Insert the coefficients \( a \), \( b \), and \( c \) into the formula.
• Compute the discriminant \( b^2 - 4ac \) to determine the type of roots.
• Simplify under the square root to find the exact values of the solutions.

For the equation \( \frac{1}{6}x^2 + x + \frac{1}{3} = 0 \), the solutions are calculated as \( x_1 = -3 + \sqrt{7} \) and \( x_2 = -3 - \sqrt{7} \). These are precise locations on the x-axis where the graph of the equation crosses, representing the real number solutions.