Problem 14 - Algebra 2 and Trigonometry

Question

Solve and check each of the equations. \(9=x(6-x)\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Rewrite the Equation

We start with the given equation: \[ 9 = x(6 - x) \]Distribute the \(x\) across the parenthesis:\[ 9 = 6x - x^2 \]Rearrange the equation to bring all terms to one side to form a standard quadratic equation:\[ x^2 - 6x + 9 = 0 \]

Step 2: Identify the Quadratic Form

The quadratic equation is now in standard form:\[ x^2 - 6x + 9 = 0 \]

Step 3: Factor the Quadratic Equation

Next, we factor the quadratic equation. Look for two numbers whose product is \(9\) and sum is \(-6\). These numbers are \(-3\) and \(-3\), so the equation factors as:\[ (x - 3)(x - 3) = 0 \]Which can be written as:\[ (x - 3)^2 = 0 \]

Step 4: Solve for x

To find \(x\), solve the equation \[ (x - 3)^2 = 0 \]This implies \(x - 3 = 0\). Solving for \(x\) gives:\[ x = 3 \]

Step 5: Verify the Solution

Substitute \(x = 3\) back into the original equation to verify:Left side: \[ 9 = x(6-x) \] becomes \[ 9 = 3(6 - 3) = 3 \times 3 = 9 \]Both sides are equal, so \(x = 3\) is a correct solution.

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental step in solving quadratic equations. The goal is to express the quadratic equation in a product form that equates to zero. This approach simplifies the process of finding the roots or solutions of the equation.

Let's consider a quadratic equation like \[ x^2 - 6x + 9 = 0 \]. To factor it, search for two numbers whose product is the constant term (in this case, 9) and whose sum equals the coefficient of the linear term (here, -6).

• Product: 9
• Sum: -6

The numbers that satisfy these conditions are -3 and -3. Therefore, the equation can be factored as:\[ (x - 3)(x - 3) = 0 \].

Factoring transforms the quadratic expression into a simpler, recognizable pattern (products that can be set to zero), making it easier to solve.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. After factoring a quadratic equation, solving it requires using the Zero Product Property. This property tells us that if the product of two numbers is zero, at least one of the numbers must be zero.

Consider the factored form \((x - 3)(x - 3) = 0\). Applying the Zero Product Property, we solve:

• \(x - 3 = 0\)

This gives:\(x = 3\). The solution \(x = 3\) is derived from setting each factor equal to zero and solving the resulting simple linear equations.

In many cases, each factor can lead to a different solution, but in this instance, because the factors are the same, we only find one unique solution.

Verification of Solutions

Once a potential solution is found in any equation-solving process, verifying the solution ensures its correctness. Verification is a crucial step in confirming that the solution satisfies the original equation.

Take the verified solution \(x = 3\) for the equation \(9 = x(6-x)\). Substitute \(x = 3\) back into the equation to check:

• Original equation: \(9 = 3(6-3)\)
• Simplified: \(9 = 3 \times 3\)
• Result: \(9 = 9\)

Both sides of the original equation match, confirming that \(x = 3\) is indeed a correct solution. Verification provides a sense of confidence and accuracy in mathematics by validating the solution against the original problem.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential concept in solving quadratic equations. It is expressed as:\[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

In the exercise, the equation \(9 = x(6-x)\) was initially not in standard form. By distributing the \(x\) and rearranging terms, we transformed it into \(x^2 - 6x + 9 = 0\), thus putting it in standard form.

Converting to standard form is a key step because:

• It allows for easy identification of \(a\), \(b\), and \(c\), which are needed for factoring and other solving methods.
• It provides a clear structure for analysis and solution.

Understanding and converting to the standard form lays the groundwork for solving quadratic equations through various methods.