Problem 2 - College Algebra Essentials

Question

Solve each equation in Exercises \(1-14\) by factoring. $$ x^{2}-13 x+36=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 - 13x + 36 = 0\) by factoring are \(x = 9\) and \(x = 4\).

Step 1: Set Up the Problem

We have the quadratic equation \(x^2 - 13x + 36 = 0\). This is in the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -13\), and \(c = 36\).

Step 2: Factoring the Quadratic

The quadratic expression \((x^2 - 13x + 36)\) can be factored by finding two numbers that multiply to \(36\) and add up to \(-13\). These numbers are \(-9\) and \(-4\). Therefore, the factored form of the expression is \((x - 9)(x - 4) = 0\).

Step 3: Solving for x

A product of factors equals zero if and only if at least one of the factors equals zero. Therefore, we can set each factor equal to zero and solve for \(x\): \n1) \(x - 9 = 0\) gives \(x = 9\). \n2) \(x - 4 = 0\) gives \(x = 4\).

Key Concepts

Quadratic Equations

Quadratic equations form a cornerstone in mathematics. These equations are typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). When compared to linear equations, quadratics involve the square of the variable, \(x\).
This introduces a parabolic graph shape. Quadratics can have zero, one, or two real solutions because the graph can intersect the x-axis at these possible points.
Understanding a quadratic equation means knowing the role each component plays:

• \('a'\) - Determines the parabola's direction (upwards if positive, downwards if negative)
• \('b'\) - Affects the position of the vertex horizontally
• \('c'\) - Influences where the parabola crosses the y-axis

The ultimate goal is to identify the values of \(x\) that satisfy the equation, which are also known as the roots of the equation.

Factoring

Factoring is a pivotal technique in algebra used to simplify expressions and solve equations. To factor a quadratic, like \(x^2 - 13x + 36\), you seek two numbers that both multiply to \(c\) (here, 36) and sum to \(b\) (here, -13).
The process often includes:

• Identifying the product-sum pair. In our example, \(-9\) and \(-4\) multiply to \(36\) and add to \(-13\).
• Expressing the quadratic as a product of binomials. The equation becomes \((x-9)(x-4)\).

Factoring transforms a quadratic into a form that is more manageable to work with. Not every quadratic can be factored easily, especially if the roots are not whole numbers, but this approach remains one of the first steps attempted in solving quadratics.

Solving Equations

Solving equations involves finding the value(s) of the variable that makes the equation true. For factored quadratic equations, such as \((x-9)(x-4)=0\), the key principle is that if a product equals zero, then at least one factor must be zero.
This principle allows us to set each binomial factor equal to zero to solve for \(x\):

• For \(x-9=0\), solving gives \(x=9\).
• For \(x-4=0\), solving gives \(x=4\).

Thus, \(x=9\) and \(x=4\) are the solutions. These solutions represent the x-values where the parabola represented by \(x^2 - 13x + 36 = 0\) intersects the x-axis.
Factoring simplifies the process of finding these critical points in comparison to other methods like completing the square or using the quadratic formula.