Question

Solve the equation. $$(x+4)(x-8)=0$$

Short answer

The solutions to the equation are \(x = -4\) and \(x = 8\)

Step-by-step solution

Apply the Zero-Product Property

Given this equation \((x+4)(x-8)=0\), one can use the zero-property rule which states if \((x+4)(x-8) = 0\), then either \(x+4 = 0\) or \(x-8 = 0\)

Solve Each Equation Individually

Solving \(x+4 = 0\) for \(x\), one gets \(x = -4\). Similarly, when solving \(x-8 = 0\) for \(x\), one gets \(x = 8\)

State the Solution Set

The solutions to the equation are the roots for the given polynomial. The roots are the values of \(x\) for which the polynomial equals zero. Therefore, the solutions are \(x = -4, 8\)

Key concepts

Zero-Product Property

The zero-product property is essential when solving quadratic equations and other polynomial equations. It's based on a simple but powerful idea: if the product of two expressions is zero, then at least one of the expressions must also be zero. In algebraic terms, if ab = 0, then either a = 0 or b = 0, or both.

This property is particularly useful because it allows us to take a complex equation and break it down into simpler parts. For instance, consider the quadratic equation (x+4)(x-8)=0. According to the zero-product property, we can conclude that either x+4=0 or x-8=0. By solving these simpler linear equations, we find the roots of the original quadratic equation. As a student becomes more familiar with this property, solving polynomial equations becomes a matter of recognizing the possible products that could equate to zero.

Quadratic Roots

Quadratic roots, also known as solutions or zeros of the quadratic equation, are the values of x that make the equation equal to zero. The standard form of a quadratic equation is ax^2 + bx + c = 0, and the solutions to this equation are called its roots.

These roots can be found by various methods including factoring, completing the square, or using the quadratic formula. In the given exercise (x+4)(x-8)=0, we applied the zero-product property to find the quadratic roots, where x = -4 and x = 8. Understanding the concept of roots is crucial as it helps in graphing quadratic functions and analyzing the points where the graph intersects the x-axis.

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A quadratic equation is a special type of polynomial equation where the highest degree is two. The general form of a polynomial equation is an*x^n + an-1*x^(n-1) + ... + a1*x + a0 = 0, where the coefficients a0, a1, ..., an are real numbers, and n is a non-negative integer representing the degree of the equation.

When solving polynomial equations, the goal is to find all the values (roots) for which the polynomial equals zero. Polynomials of higher degrees might require more advanced methods, like synthetic division or using the Rational Root Theorem, to solve. In the case of quadratic equations, however, the zero-product property, factoring, the quadratic formula, and graphing are the most common methods used to find the roots.