In Exercises 93–100, factor completely.x² − 6/25 + 1/5 x

Verified step by step guidance

1. Step 1: Begin by rewriting the expression in a standard quadratic form: \( x^2 + \frac{1}{5}x - \frac{6}{25} \).
2. Step 2: Identify the coefficients: \( a = 1 \), \( b = \frac{1}{5} \), and \( c = -\frac{6}{25} \).
3. Step 3: Use the method of factoring by grouping or the quadratic formula to find the roots of the equation.
4. Step 4: If using factoring by grouping, find two numbers that multiply to \( a \times c = 1 \times -\frac{6}{25} = -\frac{6}{25} \) and add to \( b = \frac{1}{5} \).
5. Step 5: Rewrite the middle term using the two numbers found in Step 4, and factor by grouping to express the quadratic as a product of two binomials.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting them as a product of two binomials. This process is essential for simplifying expressions and solving equations. The standard form of a quadratic is ax² + bx + c, and the goal is to express it in the form (px + q)(rx + s). Recognizing patterns, such as perfect squares or the difference of squares, can aid in this process.

Common Denominators

In the given expression, the terms involve fractions, which require a common denominator for simplification. A common denominator allows us to combine or manipulate fractions effectively. In this case, the denominators are 25 and 5, and finding a common denominator helps in rewriting the expression in a more manageable form, facilitating the factoring process.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful for factoring and solving quadratics. By rearranging the expression and adding/subtracting the necessary constant, we can express it in the form (x - p)² = q, which simplifies the factoring process and aids in finding the roots of the equation.