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A quadratic equation is a type of polynomial equation of degree 2. It generally has the form \[ax^2 + bx + c = 0\], where a, b, and c are constants, and x represents the variable. In our example, the equation is \[6x^2 + 2x - 20 = 0\]. The key characteristic of a quadratic equation is the presence of the term ax^2, which makes it a second-degree polynomial. Understanding the basic structure is essential to applying the quadratic formula for solving.

Coefficients are the numerical values that multiply the variable terms in a polynomial equation. In the quadratic equation \[6x^2 + 2x - 20 = 0\], the coefficients are:

• a (coefficient of x²) = 6
• b (coefficient of x) = 2
• c (constant term) = -20

These coefficients are crucial because you substitute them into the quadratic formula. For the formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], you need the exact values of a, b, and c to solve for x.

The discriminant is part of the quadratic formula inside the square root. It is given by the expression \[b^2 - 4ac\]. The discriminant tells you about the nature of the roots of the quadratic equation. For our example,

• b = 2
• a = 6
• c = -20

Calculating the discriminant: \[2^2 - 4 \cdot 6 \cdot (-20) = 4 + 480 = 484\]

• If the discriminant is positive, the equation has two real and distinct roots.
• If it is zero, the equation has exactly one real root (or a repeated root).
• If it is negative, the equation has no real roots, but two complex roots.

In our case, because the discriminant is 484, which is a positive number, the quadratic equation has two distinct real roots.

The square root symbol (√) is used in the quadratic formula to solve for the variable x. In our equation, once we simplified the discriminant (\[b^2 - 4ac = 484\]), we needed to find the square root of 484. The square root of a number y is a value which, when multiplied by itself, gives y. For instance, \[\sqrt{484} = 22\], because \[22 \cdot 22 = 484\]. So, when you substitute this back into the quadratic formula:\[x = \frac{-2 \pm 22}{12}\]. This gives two possible values for x by evaluating the plus and minus options:

• \[x = \frac{-2 + 22}{12} = \frac{20}{12} = \frac{5}{3}\]
• \[x = \frac{-2 - 22}{12} = \frac{-24}{12} = -2\]

Thus, the solutions obtained are \[x = \frac{5}{3}\] and \[x = -2\]. Understanding how to handle square roots is critical in solving quadratic equations using the quadratic formula.