Problem 54 - Algebra A Combined Function

Question

Solve each equation. $$ 3 x^{2}-6 x-9=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -1\).

Step 1: Identify and Write Down the Equation

We start by identifying the given quadratic equation: \(3x^2 - 6x - 9 = 0\). This is a standard form quadratic equation \(ax^2 + bx + c = 0\) with coefficients \(a = 3\), \(b = -6\), and \(c = -9\).

Step 2: Simplify the Equation

Notice that each term in the equation is divisible by 3. Divide the entire equation by 3 to simplify: \(x^2 - 2x - 3 = 0\).

Step 3: Factorize the Quadratic Equation

We look for two numbers that multiply to -3 (the constant term, \(c\)) and add to -2 (the coefficient of \(x\), \(b\)). These numbers are -3 and 1. Thus, the equation can be factorized as: \((x - 3)(x + 1) = 0\).

Step 4: Solve the Factorized Equation

Set each factor equal to zero to solve for \(x\): \(x - 3 = 0\) or \(x + 1 = 0\). This gives the solutions \(x = 3\) and \(x = -1\).

Step 5: Verify the Solutions

Substitute \(x = 3\) and \(x = -1\) back into the original equation to verify: - For \(x = 3\): \(3(3)^2 - 6(3) - 9 = 0\), which simplifies to 0. - For \(x = -1\): \(3(-1)^2 - 6(-1) - 9 = 0\), which also simplifies to 0. Both solutions are verified.

Key Concepts

Factoring Quadratic Equations

When faced with a quadratic equation, like \(3x^2 - 6x - 9 = 0\), factoring is one of the strategies you can use to solve it. Initially, you might observe that every term in this equation is divisible by 3. To simplify it, you divide the entire equation by 3, resulting in \(x^2 - 2x - 3 = 0\). Breaking down the equation into simpler parts involves expressing it as a product of two binomials.

To factor \(x^2 - 2x - 3\), consider two numbers that multiply to get -3 (the constant term) and add up to -2 (the coefficient of \(x\)). Negative three and one are the perfect pair for this case because \(-3 \cdot 1 = -3\) and \(-3 + 1 = -2\). This allows you to express the equation as \((x - 3)(x + 1) = 0\). Factoring transforms the quadratic equation into a much simpler expression that is easier to work with.

Solving Quadratic Equations

Once you've factorized the quadratic equation into \((x - 3)(x + 1) = 0\), solving it becomes straightforward. The principle of solving quadratic equations by factoring is based on the Zero Product Property. This principle states that if the product of two numbers is zero, then at least one of the numbers must be zero.

Apply this property by setting each factor equal to zero:

• For \(x - 3 = 0\), solving gives \(x = 3\).
• For \(x + 1 = 0\), solving gives \(x = -1\).

So, the solutions to the quadratic equation \(3x^2 - 6x - 9 = 0\) are \(x = 3\) and \(x = -1\). These solutions make the equation true, as substituting them back into the factorized form will yield zero.

Verifying Solutions of Equations

After solving a quadratic equation by factoring, it is crucial to verify the solutions. Verification involves substituting the solutions back into the original equation \(3x^2 - 6x - 9 = 0\) to ensure that they satisfy it.

Check each solution:

• For \(x = 3\): Substitute into the equation, \(3(3)^2 - 6(3) - 9 = 0\). Simplifying, you get \(27 - 18 - 9 = 0\), which simplifies to 0.
• For \(x = -1\): Substitute, \(3(-1)^2 - 6(-1) - 9 = 0\). Simplifying, you get \(3 + 6 - 9 = 0\), which also simplifies to 0.

Both solutions verify as correct, confirming that our calculations were accurate. Verifying solutions is an essential step in equation solving, providing confidence that no errors were made along the way.