Problem 79 - Introductory Algebra for College Students

Question

Factor completely. $$x^{2}+0.5 x+0.06$$

Step-by-Step Solution

Answer

The complete factorization of the equation \(x^{2} + 0.5x + 0.06\) is \((x + 0.2)(x + 0.3)\).

Step 1: Identify the quadratic equation

The given equation \(x^{2} + 0.5x + 0.06\) is a quadratic equation in the standard form \(ax^{2} + bx + c\), where \(a = 1\), \(b =0.5\), and \(c =0.06\).

Step 2: Determine the co-efficients

From the equation, the co-efficients are \(a=1\), \(b=0.5\), and \(c=0.06\).

Step 3: Find the roots of the equation

The roots of a quadratic equation are given by \((-b±sqrt{b^{2}-4ac})/(2a)\). In this case, they are \((-0.5±sqrt{(0.5)^{2}-4*1*0.06})/2*1\). Hence, the roots are \(-0.2\) and \(-0.3\). These roots are the values of p and q.

Step 4: Write the equation in factored form

Given the roots determined in the previous step, we now write the equation in the factored form \((x - p)(x - q)\) as \((x + 0.2)(x + 0.3)\).

Key Concepts

Quadratic Equations

Quadratic equations are essential components of algebra. They help solve problems involving areas, projectiles, and numerous real-world situations. A quadratic equation has the general form:

• \[ ax^2 + bx + c = 0 \]

Here, \( a \), \( b \), and \( c \) are coefficients. The leading term, \( ax^2 \), makes it a quadratic. The solutions or "roots" of the equation, which are the values of \( x \) making the equation true, can be found using multiple techniques. These techniques allow us to find where the parabola defined by the quadratic intersects the x-axis.

Factoring Techniques

Factoring is a technique used to rewrite quadratic equations into a product of simpler polynomials. The process involves expressing the equation \( x^2 + 0.5x + 0.06 \) as a product of two binomials:

• \( (x + p)(x + q) \)

To factor, one must find two numbers that multiply to the constant term \( c \), and add up to the linear coefficient \( b \). In our exercise, identify the roots, which solve the equation by setting it equal to zero. Once the roots are determined, they can be used to express the quadratic in factored form. Factoring is often a preferred method because it provides insight into the structure of the equation and simplifies solving.

Solving Quadratics

To solve quadratics, a variety of methods can be employed like factoring, completing the square, and using the quadratic formula. The quadratic in the exercise, \( x^2 + 0.5x + 0.06 \), is solved by factoring:

• First, find the roots using the formula \((-b \pm \sqrt{b^2 - 4ac})/(2a)\). This gives us \(-0.2\) and \(-0.3\).
• Write the equation in its factored form as \((x + 0.2)(x + 0.3)\).

By equating each factor to zero, \((x + 0.2) = 0\) or \((x + 0.3) = 0\), the values \( x = -0.2 \) and \( x = -0.3 \) are confirmed as roots. Solving quadratics by factoring is efficient and effective when the polynomial easily breaks down into linear factors.