Problem 83 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(y^{2}-6 y-7\)

Step-by-Step Solution

Answer

(y - 7)(y + 1)

Step 1: Identify coefficients

Identify the coefficients of the trinomial. In this case, the trinomial is of the form \(ax^2 + bx + c\) with \(a=1\), \(b=-6\), and \(c=-7\).

Step 2: Find two numbers that multiply to c and add to b

Find two numbers that multiply to \(c = -7\) and add to \(b = -6\). The numbers are \(-7\) and \(1\) because \((-7) \times 1 = -7\) and \((-7) + 1 = -6\).

Step 3: Write the factors

Using the two numbers found in the previous step, write the factors of the trinomial. The trinomial can be factored as:\(y^2 - 6y - 7 = (y - 7)(y + 1)\)

Key Concepts

Factoring Trinomials

Factoring trinomials is a key skill in algebra. A trinomial is an algebraic expression with three terms. To factor a trinomial, we look for two numbers that multiply to the constant term (the third term) and add to the coefficient of the middle term. This helps in breaking down complex quadratic expressions into simpler binomials.
Let's break it down with an example:
1. Identify the coefficients: For the trinomial \(y^{2}-6y-7\), the coefficients are \(a = 1\), \(b = -6\), and \(c = -7\).
2. Find two numbers that multiply to \(c\) and add to \(b\): In this case, \(-7 \times 1 = -7\) and \(-7 + 1 = -6\).
3. Write the factors: \(y^{2} - 6y - 7 = (y - 7)(y + 1)\).
By practicing these steps, you will become more comfortable with factoring trinomials and solving quadratic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent real-world scenarios or mathematical relationships.
Here’s a quick breakdown of essential components:

• **Variables**: Symbols (like \(y\)) that represent unknown values.
• **Constants**: Fixed numbers (like \(-7 & 1\) in the trinomial example).
• **Coefficients**: Numbers multiplying the variables (like -6, which is the coefficient of y in the example).

When working with algebraic expressions:

• Combine like terms: Group terms with the same variables.
• Use the distributive property: Multiply terms outside the parentheses with terms inside the parentheses.
• Factor: Rewrite expressions as the product of simpler terms.

Mastering these concepts will make solving algebra problems much easier. With practice, you will intuitively recognize patterns and efficiently simplify expressions.

Quadratic Equations

Quadratic equations are algebraic equations of the second degree, generally in the form \(ax^{2} + bx + c = 0\). They are key in algebra and appear frequently in various real-world applications.
Here’s how we typically solve them:

• **Factoring**: Rewrite the equation as a product of two binomials. For example, \(y^{2} - 6y - 7 = (y - 7)(y + 1)\).
• **Completing the Square**: Rewrite the equation in the form \((x - p)^{2} = q\) to solve for \(x\).
• **Quadratic Formula**: Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

To factor a quadratic equation:

• 1. Identify \(a, b, \& c\).
• 2. Find numbers that satisfy the product-sum condition.
• 3. Rewrite the quadratic as a product of binomials.

By mastering these methods, you can solve any quadratic equation efficiently and accurately. This foundational algebra skill is essential for higher-level mathematics and various applications in science and engineering.