91Ó°ÊÓ

A "quadratic equation" is a type of polynomial equation of the second degree. It takes the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) cannot be zero. This restriction ensures that the term \( x^2 \) is present, making the equation "quadratic"- The highest power of the variable \( x \) is 2, which makes these equations particularly interesting as they typically have two solutions.
- These solutions correspond to the points where a parabola, representing the quadratic function, intersects the x-axis.
When solving a system of equations that includes a quadratic equation, one method is to substitute the expression for one variable from the linear equation into the quadratic equation. By doing so, you derive a single equation with one variable: the quadratic equation. This allows using typical quadratic solving techniques such as factoring, completing the square, or the quadratic formula.

The "quadratic formula" is a powerful tool for finding the solutions to any quadratic equation. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a breakdown of what the formula components stand for:

• \( b^2 - 4ac \) is called the "discriminant." It tells us the nature of the roots (solutions) of the quadratic equation.
• An important aspect of the discriminant is its value:
  • If \( b^2 - 4ac > 0 \), the quadratic has two distinct real roots.
  • If \( b^2 - 4ac = 0 \), then there is exactly one real root, meaning the parabola touches the x-axis at just one point.
  • If \( b^2 - 4ac < 0 \), the quadratic has no real roots, as the parabola does not intersect the x-axis. It results in complex solutions.

In applying the quadratic formula, it is crucial to substitute \( a \), \( b \), and \( c \) correctly from the equation \( ax^2 + bx + c = 0 \). This step ensures accurate calculation of both potential roots.

"Simplifying radicals" is a technique used to make expressions involving square roots easier to work with. When you simplify a radical, you express the square root in its simplest form.- For example, \( \sqrt{80} \) can be broken down into its prime factors: \( \sqrt{16 \times 5} \).- Here we recognize that \( 16 \) is a perfect square, so \( \sqrt{16} = 4 \).
- Thus, \( \sqrt{80} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5} \).To simplify a radical:

• Identify if the number under the square root sign possesses any perfect square factors.
• Extract the square root of those perfect square factors.
• Multiply the results to express the radical in its simplest form.

Working with simplified radicals can make the process of solving equations faster and less prone to errors. Always simplify radicals in your final answers to provide a cleaner solution.