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The quadratic formula is a key concept in solving quadratic equations. It is used to find the roots of any quadratic equation, which has the general form:
\(ax^2 + bx + c = 0\)
The formula is:
\(\text{ x = } \frac{-b \text{±} \text{ √(b²-4ac)}}{2a}\)
This formula comes in handy when other methods, like factoring, are too difficult or impossible to apply. Understanding each part of the formula is essential:

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

The discriminant, \(\text{b²-4ac}\), inside the square root sign (radical) determines the nature of the roots:

• If \(\text{b²-4ac > 0}\), there are two real and distinct roots
• If \(\text{b²-4ac = 0}\), there is one real root (also known as a repeated root)
• If \(\text{b²-4ac < 0}\), there are no real roots, only complex roots

Understanding the quadratic formula will enable you to solve any quadratic equation efficiently.

The substitution method is a powerful tool used in algebra to solve equations by replacing variables with given values. This eliminates unknowns and simplifies calculations.
Here is how to effectively use the substitution method:

• Identify the given values for the variables in the equation. In our problem, we are given \(a=6\), \(b=7\), and \(c=2\)
• Write down the expression with the variables
• Substitute the given values into the expression. As seen in the exercise, we replace \(a\), \(b\), and \(c\) with their respective values in \(\text{√(b²-4ac)}\): \(\text{√(7² - 4*6*2)}\)
• Simplify the expression by carrying out the arithmetic operations:
  • Calculate \(b²\): \(\text{b² = (7)² = 49}\)
  • Find \(4ac\): \(4*6*2 = 48\)
  • Subtract the result of \(4ac\) from \(b²\): \(\text{49-48 = 1}\)
  • Finally, compute the square root: \(\text{√1 = 1}\)

The substitution method helps you handle various algebra problems systematically and avoid errors.

Radicals, often referred to as roots, are expressions that involve the root of a number or variable. They are a fundamental part of algebra.
The most common radical is the square root, denoted by the √ symbol. The expression √x represents the number which, when multiplied by itself, gives x.
For example:

• √49 = 7, because 7*7 = 49
• √1 = 1, because 1*1 = 1

In our problem, \(\text{√(b²-4ac)}\) involves finding the square root of a difference, which is often encountered in quadratic equations. Simplifying the expression inside the radical first is crucial:

• Perform all multiplications and additions/subtractions inside the radical first.
• Calculate the square root in the last step, as seen in our example: \(\text{√1 = 1}\)

Understanding how to handle radicals is important for solving a variety of algebraic problems, including those involving the quadratic formula and simplifying expressions.