Problem 75 - Elementary Algebra

Question

Use the \(a c\) method to factor \(x^{2}+0 x-36\).

Step-by-Step Solution

Answer

The factored form is \((x + 6)(x - 6)\).

Step 1 - Identify coefficients

Identify the coefficients of the quadratic equation in standard form. For the equation \(x^{2} + 0x - 36\), the coefficients are: \(a = 1\), \(b = 0\), and \(c = -36\).

Step 2 - Find the product of \(a\) and \(c\)

Multiply the coefficients \(a\) and \(c\) to find their product. In this case: \(a \cdot c = 1 \cdot (-36) = -36\).

Step 3 - Find factors of the product

Find two numbers that multiply to \(-36\) and add to \(0\) (the coefficient of \(x\)). These numbers are \(-6\) and \(6\) as \((-6) \cdot 6 = -36\) and \((-6) + 6 = 0\).

Step 4 - Rewrite the middle term

Rewrite the middle term using the factors found in Step 3. So, \(x^{2} + 0x - 36\) becomes \(x^{2} - 6x + 6x - 36\).

Step 5 - Group and factor

Group the terms in pairs and factor out the common factors. For \(x^2 - 6x + 6x - 36\): \(x(x - 6) + 6(x - 6)\).

Step 6 - Factor by grouping

Notice that \(x - 6\) is a common factor. Factor it out: \((x + 6)(x - 6)\).

Key Concepts

ac method

The 'ac method' is a technique used for factoring quadratic equations. It is especially helpful when the quadratic trinomial cannot be factored easily by inspection. The method involves four primary steps: multiplying the coefficients of the first and last terms (in the form of ax^2 + bx + c), finding two numbers that multiply to this product and add to the middle coefficient, rewriting the middle term using these numbers, and finally grouping and factoring the rewritten equation. This method simplifies solving otherwise tricky quadratic equations.

quadratic equation coefficients

Quadratic equations can typically be written in the standard form: ax^2 + bx + c. The coefficients 'a', 'b', and 'c' are the numerical factors that multiply with the variable terms. Identifying these coefficients correctly is crucial in methods like the 'ac method'.

• In the quadratic equation x^2 + 0x - 36:
• a = 1 (the coefficient of x^2)
• b = 0 (the coefficient of x)
• c = -36 (the constant term)

This identification aids in determining the next steps in factoring the equation correctly.

factor by grouping

Factoring by grouping is a valuable skill in algebra. After rewriting the middle term using the two numbers from the 'ac method', the quadratic equation can be expressed in four terms. These terms can be grouped in pairs and factored separately. For instance:
For x^2 - 6x + 6x - 36:

• Group terms into pairs: (x^2 - 6x) and (6x - 36)
• Factor out the greatest common factor in each pair: x(x - 6) + 6(x - 6)
• Observe the common binomial factor: (x - 6)
• Factor out the common binomial to get the fully factored form: (x + 6)(x - 6)

Grouping simplifies the quadratic expression into a product of binomials, making it easy to solve.

rewriting middle term

One critical step in the 'ac method' is rewriting the middle term. This is necessary to simplify the quadratic expression into a form that can be factored by grouping. For x^2 + 0x - 36, we find two numbers that multiply to 'ac' (= -36) and add to 'b' (= 0), which are -6 and 6.

• Replace the middle term with two terms using these factors: x^2 - 6x + 6x - 36
• This step transforms the equation, making it possible to factor by grouping

Rewriting the middle term is a strategic maneuver, breaking down the problem into manageable parts and setting up for easy factoring.