91Ó°ÊÓ

Quadratic equations are fundamental in algebra and appear in many different contexts. A quadratic equation usually takes the form \(ax^2 + bx + c = 0\). The goal when solving these equations is to find the values of \(x\) that make the equation true. Quadratic equations can have either one or two solutions. These solutions are sometimes referred to as the

• roots
• zeroes

of the equation.
In our original exercise, the equation given is already simplified into a product form:

• \((y+9)(y-2)=0\)

Using the zero-product property, we can easily identify the solutions by determining the values of \(y\) for which each factor equals zero.

Factoring quadratic expressions is the process of rewriting the equation in a way that is easier to solve, specifically so it can be set to zero and solved using the zero-product property. A quadratic expression can often be factored into the product of two binomials. For example, a quadratic like \( y^2 + 7y + 10 \) would be factored into

• \((y + 5)(y + 2)\)

Once the equation is factored, it is transformed into a product form. This step makes identifying possible solutions straightforward, as you can apply the zero-product property. Each factor set equal to zero may result in a different solution. For our initial equation

• \((y+9)(y-2)=0\)

we see it is already factored, which shows the importance of simplifying expressions to solve equations.

Equation solutions are the values that satisfy the equation. When dealing with quadratic equations, these solutions represent the points where the graph of the equation intersects *or* crosses the x-axis. This intersection is known as the ``x-intercept.'' In essence, finding these solutions is about pinpointing the specific value(s) of \(y\) that make the equation true. In our exercise, we derived two solutions for the equation

• \(y = -9\)
• \(y = 2\)

These solutions show us the specific points where the expression becomes zero. By solving each factor of the equation separately, we confirmed the solutions satisfy the original equation. Remember, when using the zero-product, the solutions are immediately visible when solving the factored form.