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The **general form** of a quadratic equation is crucial for solving and understanding quadratic functions. It is typically expressed as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are real numbers and \(a eq 0\). The coefficient \(a\) must be non-zero; otherwise, the equation would not remain quadratic. This standard format allows us to apply various techniques to find the roots or solutions of the quadratic equation.

When dealing with word problems or real-life applications, identifying the quadratic equation in general form can simplify the solving process. You can use techniques like completing the square or the quadratic formula, which directly work with this form. Furthermore, it aids in graphing the function since commonly used graphing tools require the equation in this format. Knowing how to transition from different forms of quadratic expressions into the general form is an essential skill for anyone studying algebra.

Finding the **roots of a quadratic equation** is a key step in solving these problems because the roots (or solutions) indicate where the graph of the quadratic function intersects the x-axis. A quadratic equation can have either:

• Two distinct real roots,
• One real root (repeated root), or
• No real roots (when the roots are complex numbers).

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), is a powerful tool for finding the roots from the general form. It calculates the roots by finding where the parabola cuts the x-axis. In our exercise, we already have the roots, which are -3 and 5. To build the quadratic equation from these roots, we use the fact that if \(p\) and \(q\) are roots, the equation can be represented as \((x-p)(x-q)=0\). By using this method, you directly correlate the algebraic expression with its graphical interpretation.

**Expanding binomials** is one of the core techniques in algebra that allows us to transform products of binomials into a sum of terms. This is especially useful when starting with the root form of a quadratic equation and turning it into the general form.

• Consider the expression \((x + 3)(x - 5)\), derived from our roots of -3 and 5.
• When expanded, it uses the distributive property, sometimes referred to as the FOIL method (First, Outer, Inner, Last).

For this example:

• First: Multiply the first terms: \(x \times x = x^2\)
• Outer: Multiply the outer terms: \(x \times -5 = -5x\)
• Inner: Multiply the inner terms: \(3 \times x = 3x\)
• Last: Multiply the last terms: \(3 \times -5 = -15\)

Gathering these together, the expanded form becomes \(x^2 - 5x + 3x - 15\), which simplifies to \(x^2 - 2x - 15\). This expanded form is crucial to rewriting the equation into the general form noted as \(ax^2 + bx + c = 0\). Understanding how to expand binomials enables students to convert from the factored form to the standard general form smoothly.