Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by factoring. Check your solutions in the original equation. $$9 x^{2}-21 x=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x=0\) and \(x=7/3\).

Step 1: Identify the common factor

In the equation \(9x^{2}-21x=0\), the common factor is 3x. You can factor this out, which simplifies the equation to \(3x(3x-7)=0\).

Step 2: Apply the Zero Product Property

The Zero Product Property states that for any real numbers a and b, if ab = 0, then either a = 0, or b = 0, or both. Apply this property to \(3x(3x-7)=0\), which gives two equations: \(3x=0\) and \(3x-7=0\).

Step 3: Solve each equation

Now you can solve each of the equations for the variable. For \(3x=0\), divide both sides by 3 to get \(x=0\). For \(3x-7=0\), add 7 to both sides first to get \(3x=7\), then divide both sides by 3 to get \(x=7/3 \) or \(x=2.33\) when rounded to two decimal places.

Step 4: Check the solutions in the original equation

Substitute the obtained solutions \(x=0\) and \(x=7/3\) into the original equation and verify if the equation is satisfied. Upon substitution, both the solutions satisfy the original equation and hence are the correct solutions.

Key Concepts

Factoring

Factoring is a method used to simplify equations, making it easier to find solutions. When you factor an expression, you look for the greatest common factor that divides each term. For the equation \(9x^{2} - 21x = 0\), you can see that both terms have a common factor of \(3x\). Here's how you factor it out:

• Identify the common factor: In this equation, it's \(3x\).
• Rewrite the expression: When you take \(3x\) out of each term, the equation becomes \(3x(3x - 7) = 0\).

By factoring, you've transformed the quadratic equation into a product of two expressions. This is a crucial step that simplifies solving the equation further.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that helps solve equations of factored expressions. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. In the context of solving quadratic equations, this property is very useful.
When you apply this property to the factored equation \(3x(3x - 7) = 0\), you're essentially setting each factor equal to zero:

• Set \(3x = 0\)
• Set \(3x - 7 = 0\)

By solving these simple equations separately, you can determine the possible values of \(x\) that satisfy the original equation. This step takes advantage of the Zero Product Property’s simplicity for solving quadratic equations.

Equation Solving

Equation solving involves isolating the variable to find its value. After applying the Zero Product Property, you get two equations to solve from \(3x(3x - 7) = 0\):

• \(3x = 0\)
• \(3x - 7 = 0\)

Start with \(3x = 0\):

• Divide both sides by 3 to isolate \(x\).
• This gives \(x = 0\).

Next, solve \(3x - 7 = 0\):

• Add 7 to both sides to get \(3x = 7\).
• Then divide by 3 to isolate \(x\), which gives \(x = \frac{7}{3}\).

These steps help find the specific solutions of the quadratic equation, ensuring each possibility from factoring is considered.

Polynomial Expressions

A polynomial expression is a mathematical expression involving a sum of powers of variables with constant coefficients. Quadratic equations like \(9x^{2} - 21x = 0\) are specific cases of polynomial expressions, typically taking the form \(ax^{2} + bx + c = 0\). In this instance:

• The term \(9x^{2}\) represents the quadratic part.
• The term \(-21x\) is linear, influencing the slope.
• There's no explicit constant term \(c\) here, so it's like \(+0\).

Understanding these components is vital in recognizing what methods will best simplify and solve the equation, such as factoring, using the Zero Product Property, or applying other algebraic techniques. These equations form a foundational aspect of algebra, serving as a gateway to more advanced topics in mathematics.