Problem 19 - Algebra A Combined Function

Question

Solve. $$ x^{2}-13 x+36=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = 9\).

Step 1: Identify the Quadratic Equation Structure

The equation given is \(x^2 - 13x + 36 = 0\). It is in the standard form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -13\), and \(c = 36\).

Step 2: Determine Factors of the Constant Term

To solve by factoring, look for two numbers that multiply to the constant term \(c = 36\) and add to the coefficient of the linear term \(b = -13\).

Step 3: Find the Pair of Numbers

The numbers that multiply to 36 and add to -13 are -4 and -9. That is because \((-4) \times (-9) = 36\) and \((-4) + (-9) = -13\).

Step 4: Use the Factors to Break Down the Quadratic

Rewrite the quadratic equation using the factors found: \((x - 4)(x - 9) = 0\).

Step 5: Solve for the Roots

Set each factor equal to zero: \(x - 4 = 0\) and \(x - 9 = 0\). Solve these equations to get \(x = 4\) and \(x = 9\).

Key Concepts

Factoring Method

The factoring method is one of the key techniques to solve quadratic equations. It transforms a quadratic expression into a product of two simpler expressions. This method involves writing the quadratic equation in the form of \( ax^2 + bx + c = (x + p)(x + q) \). Here, the challenge is to find two numbers \( p \) and \( q \) that fulfill two conditions:

• Their product must equal the constant term \( c \).
• Their sum must equal the linear coefficient \( b \).

For example, in the equation \( x^2 - 13x + 36 = 0 \), you need numbers that multiply to 36 and add to -13. These numbers are -4 and -9. So, the equation can be rearranged to \( (x - 4)(x - 9) = 0 \).
Once the equation is factored, each separate factor can be set to zero, giving simple equations to solve. The solutions to these simpler equations are the roots of the original quadratic equation.

Roots of Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation. Finding these roots essentially means solving the equation to find where it equals zero.
For the factored equation \( (x - 4)(x - 9) = 0 \), each factor represents a potential root. You set each to zero and solve:

• \( x - 4 = 0 \) leads to the root \( x = 4 \)
• \( x - 9 = 0 \) gives the root \( x = 9 \)

Thus, the roots of the equation \( x^2 - 13x + 36 = 0 \) are \( x = 4 \) and \( x = 9 \). These roots are the points where the parabola defined by the equation crosses the x-axis in a graph. Knowing the roots helps in graphing and understanding the behavior of quadratic functions.

Standard Form

A quadratic equation is generally expressed in its standard form as \( ax^2 + bx + c = 0 \). This format makes it easy to identify the coefficients \( a \), \( b \), and \( c \), which play crucial roles in various algebraic operations. In this standard form:

• \( a \) is the coefficient of \( x^2 \), indicating the equation's "shape" or "width."
• \( b \) is the coefficient of \( x \), influencing the equation's horizontal position.
• \( c \) is the constant term, determining the y-intercept when graphed.

For the equation \( x^2 - 13x + 36 = 0 \), \( a = 1 \), \( b = -13 \), and \( c = 36 \). The standard form is not only a base for solving quadratics but also for deriving other forms, such as vertex or intercept form. Grasping this structure is vital for understanding how each component affects the parabola's graph.