What are the values for \(x\) that satisfy the equation \((x+a)(x+b)=0 ?\) F. \(-a\) and \(-b\) G. \(-a\) and \(b\) H. \(-a b\) J. \(a\) and \(-b\) K. \(a\) and \(b\)

Quadratic equations are a fundamental element of algebra that you will encounter in various forms. These are polynomials of the second degree, usually written in the form \(ax^2 + bx + c = 0\). To solve such equations, there are multiple methods, including factoring, completing the square, and using the quadratic formula.

When a quadratic equation is factorable, as in the case of \(x+a)(x+b)=0\), we use a property called the zero product property. This rule states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for \(x\) separately, \(x+a=0\) and \(x+b=0\), which gives us the solutions \(x=-a\) and \(x=-b\).

Understanding Factoring

Factoring is one of the simplest methods used to solve quadratic equations when possible. It involves breaking down the equation into a product of simpler expressions that can more readily be solved or evaluated.

An algebraic equation represents a statement of equality between two algebraic expressions. They range from simple linear equations to more complex forms such as quadratics. Mastery of solving algebraic equations is crucial for advancing in mathematics.

Challenges with algebraic equations often stem from understanding the rules and applying them correctly. For instance, always perform the same operation on both sides of the equation to maintain balance. Also, recognize when to use techniques like isolating variables, combining like terms, and dealing with fractions.

Importance of Properties in Algebra

Properties such as the distributive property, the associative property, and the commutative property often play a role in simplifying and solving equations. Additionally, understanding the permutation of terms without changing their product (commutative property) or grouping them differently (associative property) can greatly aid in solving more elaborate equations.

Preparing for the ACT Math section involves familiarizing yourself with the types and formats of questions you'll encounter. The test covers a range of topics, including algebra, geometry, and trigonometry.

For success on ACT Math, practice is essential. Working through exercises, particularly those involving algebraic equations and their solutions, can help strengthen your skills. Familiarize yourself with various problem-solving strategies such as the elimination method, substitution, and factoring.

Test-Taking Strategies

During the exam, it's also recommended to use test-taking strategies like answering the questions you know first, eliminating obviously wrong answers, and managing your time efficiently so you can revisit more challenging problems. Additionally, understanding basic properties of algebra, like the zero product property used in solving quadratic equations, will significantly improve your performance.