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

This formula allows us to find the values of x that satisfy the equation. Understanding this formula is crucial because it can solve any quadratic equation, even when factoring is difficult or impossible.

• Identify the coefficients a, b, and c in the equation.
• Plug these coefficients into the quadratic formula.
• Simplify the expression inside the square root.
• Calculate the solutions using both the \(+\) and \(-\) signs.

Solving quadratic equations involves finding the values of x that make the equation true. In this exercise, we solve the equation \(3x^2 - 7x - 20 = 0\) using the quadratic formula.

Here’s how to solve it step by step:

• First, identify the coefficients: \(a = 3\), \(b = -7\), and \(c = -20\).
• Next, apply the quadratic formula: \[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot (-20)}}{2 \cdot 3} \]
• Simplify inside the square root: \((-7)^2 = 49\) and \(4 \cdot 3 \cdot (-20) = -240\). Adding these gives \(49 + 240 = 289\).
• Substitute back into the formula: \[ x = \frac{7 \pm \sqrt{289}}{6} \]
• Since \sqrt{289} = 17\, solve for each case: \[ x = \frac{7 + 17}{6} = 4 \] and \[ x = \frac{7 - 17}{6} = -\frac{5}{3} \]

Thus, the solutions are \(x = 4\) and \(x = -\frac{5}{3}\).

A quadratic equation is a type of polynomial, specifically a second-degree polynomial because the highest power of the variable (x) is squared.

Polynomials are expressions that can have constants, variables, and exponents. They form a large part of algebra and are used to represent a wide range of problems.

Quadratic polynomials have the form:

where:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Polynomials can be solved using various methods, including factoring, completing the square, and the quadratic formula. Understanding polynomials is essential because they form the basis for more complex algebraic concepts.

When solving quadratic equations using the quadratic formula, you’ll find the roots or solutions of the equation. These roots can be real or complex numbers.

The term under the square root sign, \(b^2 - 4ac\), is called the discriminant. The discriminant helps determine the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (it’s a repeated root).
• If the discriminant is negative, there are no real roots – instead, there are two complex roots.

In our exercise, the discriminant was \(49 + 240 = 289\), which is positive. This tells us that the quadratic equation \(3x^2 - 7x - 20 = 0\) has two distinct real roots: \(x = 4\) and \(x = -\frac{5}{3}\).

Real roots are the values of x that satisfy the equation, and they can be verified by plugging them back into the original equation.