Question

Solve the equation. $$(z+2)(z+3)=0$$

Short answer

The roots of the equation are z = -2 and z = -3.

Step-by-step solution

Set each factor equal to zero

The given equation has two factors ((z+2) and (z+3)) multiplied together and set equal to zero. To find the roots of this equation, set each factor equal to zero separately. Hence, you have two equations now: \n\[ z+2 = 0 \] \n\[ z+3 = 0 \]

Solve for z

Now, simply solve each equation for z by subtracting 2 from both sides in the first equation and 3 from both sides in the second equation. You will get: \n\[ z = -2 \] \n\[ z = -3 \]

Key concepts

Factoring Polynomials

Understanding how to factor polynomials is essential for solving quadratic equations. Factoring involves breaking down a polynomial into simpler components - these components are polynomials as well, but of lesser degree, which are multiplied together to get the original polynomial. It's akin to finding what ingredients went into a recipe by tasting the finished dish.
When dealing with quadratic equations, which are polynomials of degree 2 (the highest power of the variable is 2), factoring typically looks for two binomials that multiply to give you the original quadratic. A general form for a quadratic equation is
\( ax^2 + bx + c \). In factoring, our goal is to write it as
\( (mx + n)(px + q) = 0 \), where \( m, n, p, \) and \( q \) are numbers such that when the binomials are multiplied out, they give back the original quadratic equation.
To find these numbers, we check for a pair of factors of \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) that sum up to \( b \) (the coefficient of \( x \)). This technique is very handy when the leading coefficient is 1, as in the exercise \( (z+2)(z+3)=0 \).
In the given exercise, the polynomial was already factored into \( z+2 \) and \( z+3 \), showing an immediate application of this concept.

The Zero Product Property

The zero product property is a fundamental theorem in algebra which states that if the product of two numbers is zero, then at least one of the multiplicands must be zero. This property is the backbone of solving quadratic equations by factoring.
In mathematical terms, if \( a \) and \( b \) are real numbers and \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \), or both. The simplicity of this concept belies its power, as it allows us to take a factored quadratic equation like \( (z+2)(z+3) = 0 \) and break it down into two simpler equations: \( z+2 = 0 \) or \( z+3 = 0 \).
By applying the zero product property, we can find the roots of the quadratic equation individually. From the algebraic standpoint, this property transforms a multiplicative problem into an additive one, where we're now solving for the additive inverse of each term to isolate the variable.

Roots of Equations

The roots of equations, also known as solutions or zeroes, are the values of the variable that make the equation true. For a quadratic equation, there can be two roots, one root, or no roots depending on the discriminant. In factoring, we look for roots that satisfy the equation after it has been factored into a product of binomials.
In the context of our exercise \( (z+2)(z+3)=0 \), the roots are the values of \( z \) that make each factor zero, which are \( z = -2 \) and \( z = -3 \). These solutions can be graphically interpreted as the points where the parabola represented by the quadratic equation crosses the x-axis.
Finding roots is a critical skill in mathematics, as it allows us to understand the behavior of functions and to solve various practical problems. It's an act similar to unlocking a code; once we find the right key—the roots—we can understand the deeper workings of the function in question.