Problem 16 - Basic Technical Mathematics with Calculus

Question

$$\text { Solve the given quadratic equations by factoring.}$$ $$x^{2}+30=11 x$$

Step-by-Step Solution

Answer

The solutions are \( x = 5 \) and \( x = 6 \).

Step 1: Bring the Equation to Standard Form

Start with the given equation, which is not in standard quadratic form: \[ x^2 + 30 = 11x \]Subtract \( 11x \) from both sides to have all terms on one side:\[ x^2 - 11x + 30 = 0 \] Now the equation is in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = -11 \), and \( c = 30 \).

Step 2: Identify Factors of the Constant Term

Look for two numbers that multiply to \( c = 30 \) and add up to \( b = -11 \). These numbers should satisfy:- Product: \( m \times n = 30 \)- Sum: \( m + n = -11 \)The numbers \(-5\) and \(-6\) fit this requirement since:\[ -5 \times -6 = 30 \]\[ -5 + (-6) = -11 \]

Step 3: Write the Factored Form of the Quadratic

Using the factors found, write the quadratic as a product of binomials:\[ (x - 5)(x - 6) = 0 \]

Step 4: Solve for the Variable

Set each factor equal to zero and solve for \( x \):1. \( x - 5 = 0 \) leads to \( x = 5 \)2. \( x - 6 = 0 \) leads to \( x = 6 \)

Step 5: Verify the Solution

Verify the solutions by plugging them back into the original equation. First with \( x = 5 \):\[ 5^2 + 30 = 25 + 30 = 11 \times 5 = 55 \]Since both sides are equal, \( x = 5 \) is a solution.Next with \( x = 6 \):\[ 6^2 + 30 = 36 + 30 = 11 \times 6 = 66 \]Since both sides are equal, \( x = 6 \) is also a valid solution.

Key Concepts

Standard Form of Quadratic Equation

Quadratic equations are a staple in algebra and take the form of a polynomial equation. The most recognized format is the **standard form**, which is expressed as \( ax^2 + bx + c = 0 \). In this structure:

• \( a \) is the coefficient in front of \( x^2 \) and it cannot be zero in a quadratic equation.
• \( b \) is the coefficient of \( x \), dictating the linear part of the equation.
• \( c \) is the constant term which is the standalone number.

For example, given the equation \( x^2 + 30 = 11x \), it is not initially in the standard form. To convert this, subtract \( 11x \) from both sides to achieve \( x^2 - 11x + 30 = 0 \). Here, \( a = 1 \), \( b = -11 \), and \( c = 30 \). Recognizing the standard form is crucial as it sets the stage for solving quadratic equations using various methods, including factoring.

Factored Form of Quadratic Equation

Once a quadratic equation is in its standard form, it can be transformed into its **factored form**. The factored form appears as a product of two binomials set to zero, for example, \((x - p)(x - q) = 0\). The idea here is to find two numbers, \( p \) and \( q \), that:

• Multiply to give \( a \times c \) (in most direct cases where \( a = 1 \), simply \( c \))
• Add to give \( b \)

This process hinges on your ability to identify two numbers meeting these criteria. In our exercise, the standard form was \( x^2 - 11x + 30 = 0 \). Decomposing this requires numbers that multiply to 30 and sum to -11, which are -5 and -6. Thus, it transforms into the factored form \( (x - 5)(x - 6) = 0 \). This elegant transition makes it simpler to solve the quadratic equation.

Solving Quadratic Equations

Solving quadratic equations often involves converting to factored form and then using the **Zero Product Property**. This property states that if a product of multiple factors equals zero, then at least one of the factors must be zero. When a quadratic is in factored form like \( (x - 5)(x - 6) = 0 \), it means either:

• \( x - 5 = 0 \)
• \( x - 6 = 0 \)

Hence solving these gives:

• \( x = 5 \)
• \( x = 6 \)

Verification ensures these solutions satisfy the original equation. Placing \( x = 5 \) back into the equation yields \( 5^2 + 30 = 55 \), matching \( 11 \times 5 \). Similarly, substituting \( x = 6 \) results in both sides equating to 66. Thus, both solutions are correct, demystifying the equation. This method of solving not only roots out the values but also demystifies the quadratic nature of the equation.