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The discriminant is a key part of the quadratic equation and provides valuable insights into the nature of the equation's roots. It is denoted as \(D\) and calculated using the formula \(D = b^2 - 4ac\). Here, the letters \(a\), \(b\), and \(c\) represent the coefficients from the standard form of a quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root, which means the roots are repeated.
• If the discriminant is negative, the quadratic equation has no real roots, only complex ones.

Understanding the discriminant helps in predicting the behavior of the quadratic equation's solutions even before solving them. In our exercise, the discriminant calculated was 36, which is positive, indicating two distinct real roots.

A quadratic equation is a second-degree polynomial equation in a single variable such as \(x\). Its standard form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). Quadratic equations are a cornerstone of algebra and appear frequently in various mathematical problems.

• The term \(ax^2\) is the quadratic term. It represents the parabolic part of the equation.
• The term \(bx\) is the linear term. It shifts the parabola along the x-axis.
• The term \(c\) is the constant term. It moves the parabola up or down the y-axis.

Solutions to quadratic equations can be found using several methods, including factoring, completing the square, or the quadratic formula. In our exercise, we used the quadratic formula to solve \(x^2 - 4x = 5\). This method is often preferred for its universality and simplicity.

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). It is essential because it sets a uniform way to express quadratic equations, making analysis and solution finding systematic. To use methods like the quadratic formula correctly, we must first express the quadratic in this form.Let's break down our task:- Transform the given equation \(x^2 - 4x = 5\) into standard form by setting it to equal zero: \(x^2 - 4x - 5 = 0\).- Now, we can easily identify the elements: \(a = 1\), \(b = -4\), and \(c = -5\).This process not only prepares the equation for solution techniques but also aids in evaluating it for properties like the discriminant. Recognizing and using the standard form builds a foundation for solving any quadratic equation efficiently.