91Ó°ÊÓ

Quadratic equations are one of the most fundamental topics in algebra. They are equations of degree two, typically written in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. Solving them involves finding the values of \(x\) that make the equation true. These values are also known as the roots or solutions of the equation.
Quadratic equations can have:

• Two real solutions
• One real solution (also called a repeated root)
• No real solutions (but instead, complex solutions)

Understanding how to solve these equations is crucial because they appear frequently in various branches of mathematics and applied sciences. They can be solved using different methods including factoring, completing the square, or the quadratic formula.

Factored form is an expression that breaks down a quadratic equation into the product of its linear factors. For instance, the equation \((q-4)(q-7)=0\) is already in factored form. This form is particularly useful because it allows us to apply the Zero-Product Property directly.
The Zero-Product Property states that if the product of two numbers is zero, then at least one of the numbers must be zero. Hence, from the example

• if \((q-4) = 0\) or
• if \((q-7) = 0\)

These equalities help us find the possible values of \(q\). Using factored form simplifies the process of finding the solutions to the quadratic equation, as each factor can be set to zero and solved individually using basic algebraic techniques.

Solving linear equations is a foundational skill in algebra. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In the context of our example, after applying the Zero-Product Property to the equation \((q-4)(q-7)=0\), we derived two simpler linear equations: \(q-4=0\) and \(q-7=0\).
Solving these equations involves isolating the variable on one side:

• For \(q-4=0\), adding 4 to both sides results in \(q=4\).
• For \(q-7=0\), adding 7 to both sides results in \(q=7\).

These steps showcase how to handle equations involving a single variable, often requiring only a few straightforward operations like addition or subtraction. Once you solve these linear equations, you obtain the roots of the original quadratic equation, providing a complete solution to the problem.