Problem 22 - Algebra A Combined Function

Question

Solve. $$ x^{2}-5 x+6=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = 3\).

Step 1: Recognize the Equation Type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -5\), and \(c = 6\).

Step 2: Factor the Quadratic Equation

Quadratic equations can be solved by factoring when they can be expressed as \((x - p)(x - q) = 0\). We need to find two numbers such that their product is \(6\) (the constant term) and their sum is \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) satisfy these conditions because \((-2) + (-3) = -5\) and \((-2) \times (-3) = 6\). Thus, we can factor the equation as \((x - 2)(x - 3) = 0\).

Step 3: Solve the Factored Equation

Set each factor equal to zero: \(x - 2 = 0\) or \(x - 3 = 0\). Solving these equations gives \(x = 2\) or \(x = 3\).

Step 4: Verify the Solutions

To verify our solutions, substitute \(x = 2\) and \(x = 3\) back into the original equation. For \(x = 2\), \(2^2 - 5(2) + 6 = 0\); For \(x = 3\), \(3^2 - 5(3) + 6 = 0\). Both values satisfy the equation, confirming they are correct solutions.

Key Concepts

Factoring Quadratics

Factoring is a method used to break down quadratic equations into simpler forms that are easier to solve. The equation given is a classic example of a quadratic: \(x^2 - 5x + 6 = 0\). Here, we want to express it in the form \((x - p)(x - q) = 0\). This involves finding two numbers whose product equals the constant term (6) and whose sum equals the coefficient of the linear term (-5).

In this case, the numbers -2 and -3 work perfectly because

• Their sum is -5: \((-2) + (-3) = -5\)
• Their product is 6: \((-2) \times (-3) = 6\)

Therefore, the equation \(x^2 - 5x + 6 = 0\) can be factored as \((x - 2)(x - 3) = 0\). This method of factoring allows us to move forward to solving the equation with ease.

Solving Equations

Once we have factored a quadratic equation, solving it becomes straightforward. For the equation \((x - 2)(x - 3) = 0\), the solution involves setting each factor equal to zero.

• Set \(x - 2 = 0\)
• Set \(x - 3 = 0\)

Solving these simple equations gives us two solutions: \(x = 2\) and \(x = 3\). These solutions indicate the points at which the original quadratic equation equates to zero. This is why factoring is such a powerful tool; it breaks down what seems complex into simpler, manageable parts.

Verification of Solutions

It is crucial to ensure that the solutions found are accurate, particularly in mathematical problems. For this quadratic equation, we verify by substituting the solutions back into the original equation \(x^2 - 5x + 6 = 0\).

When substituting \(x = 2\), we calculate:

• \(2^2 - 5 \times 2 + 6 = 0\)
• Results in: \(4 - 10 + 6 = 0\)

For \(x = 3\), we have:

• \(3^2 - 5 \times 3 + 6 = 0\)
• Results in: \(9 - 15 + 6 = 0\)

Both calculations confirm that substituting these values satisfies the original equation, thus verifying that the solutions \(x = 2\) and \(x = 3\) are indeed correct. This step is vital as it confirms the accuracy and courage to proceed with confidence in problem-solving.