Question

Solve the equation using any method. Explain your reasoning. \(u^2=-9 u\)

Short answer

The solutions to the equation are \(u = 0\) and \(u = -9\).

Step-by-step solution

Express the equation in standard form

Write the quadratic equation in standard form \(Ax^2 + Bx + C = 0\). This changes the given equation to \(u^2 + 9u = 0\).

Factoring the quadratic equation

The next step is to factor the quadratic expression on the left side of the equation. Factoring \(u^2 + 9u\) gives \(u(u + 9)\).

Setting factors equal to zero and solving

A product equals zero if and only if at least one of its factors is zero. Therefore, we set each factor \(u\) and \(u + 9\) equal to zero, we get \(u = 0\) and \(u = -9\).

Key concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is crucial when solving these types of problems. A quadratic equation is typically represented as \( Ax^2 + Bx + C = 0 \), where \( A \), \( B \), and \( C \) are constants with \( A \) not equal to zero. This form is useful because it enables you to apply various methods for solving quadratic equations, such as factoring, completing the square, using the quadratic formula, and graphing.

For the equation \( u^2 = -9u \), the first step to solve it by any method is to rewrite it in standard form. To do this, you must ensure that all terms are on one side of the equals sign and the other side is zero. That's how we transform the given equation into \( u^2 + 9u = 0 \) by adding \( 9u \) to both sides. This form sets the stage for factoring or applying other solution methods and is an essential skill for anyone tackling quadratic equations.

Factoring Quadratic Equations

Factoring is a powerful technique for solving quadratic equations that can be written as a product of two binomials. The goal is to break down the equation into simpler parts that are easier to solve.

To factor the standard form quadratic equation \( u^2 + 9u = 0 \), one must look for two numbers that multiply to the constant term, \( C \), and add up to the linear coefficient, \( B \). When \( A \), the coefficient of the \( x^2 \) term, is 1, as it is in our example, the process is straightforward. You're looking for factors of \( C \) that add up to \( B \). In this case, since \( C = 0 \) and \( B = 9 \), the factorization is simply \( u(u + 9) \).

Factoring requires practice to recognize patterns and identify the factors swiftly. It's a vital concept to master, as it can make solving quadratic equations much quicker and more intuitive.

Zero Product Property

The zero product property is a fundamental principle that states if the product of two numbers or expressions is zero, then at least one of the multiplicands must be zero. This property is very useful when solving quadratic equations that have been factored into binomials.

Applying the zero product property to the factored equation \( u(u + 9) = 0 \), we infer that either \( u = 0 \) or \( u + 9 = 0 \). By solving each equation separately, we eventually find that the solutions to the original quadratic equation are \( u = 0 \) and \( u = -9 \).

Understanding and applying the zero product property is an essential step after factoring the quadratic equation. It allows us to break down complex problems into simpler ones and gives us a clear pathway to find the solutions.