Problem 16 - Basic Technical Mathematics with Calculus

Question

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: Move All Terms to One Side

Start by subtracting \(11x\) from both sides of the equation: \[x^2 + 30 - 11x = 0\] Rearrange the terms to get:\[x^2 - 11x + 30 = 0\]

Step 2: Factor the Quadratic Equation

Look for two numbers whose product is 30 and whose sum is -11. These numbers are -5 and -6.Factor the quadratic equation: \[(x - 5)(x - 6) = 0\]

Step 3: Apply the Zero-Product Property

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

Step 4: Verify the Solutions

Substitute \(x = 5\) and \(x = 6\) back into the original equation to ensure they satisfy it.For \(x = 5\):\[5^2 + 30 = 11(5) \Rightarrow 25 + 30 = 55\] For \(x = 6\):\[6^2 + 30 = 11(6) \Rightarrow 36 + 30 = 66\] Both satisfy the equation, confirming they are correct solutions.

Key Concepts

Factoring

One of the core skills in solving quadratic equations by the factoring method is to express the quadratic expression in the form of a product of two binomials. Factoring is essentially the opposite of expanding an expression. The initial step involves rewriting the quadratic equation in standard form, which is given as \(ax^2 + bx + c = 0\). Consider the original equation:

• Move all terms to one side, giving \(x^2 - 11x + 30 = 0\). This rearrangement is crucial as it allows observation of the quadratic in a standard form suitable for factoring.

The next task is to find two numbers whose product equates to the constant term \(c\) and whose sum equates to the linear coefficient \(b\). For the quadratic \(x^2 - 11x + 30\), we need numbers that multiply to 30 and add up to -11. These numbers are -5 and -6.

• This leads to the expression: \((x - 5)(x - 6) = 0\). Factoring transforms the quadratic into a simpler expression and sets up the problem to utilize the zero-product property as the next step.

Zero-Product Property

The zero-product property is a foundational algebra concept, imperative when solving factored quadratic equations. It states that if the product of two numbers is zero, at least one of the numbers must be zero. For our factored quadratic equation \((x - 5)(x - 6) = 0\), the zero-product property allows us to set each factor to zero:

• \(x - 5 = 0\) leads to \(x = 5\)
• \(x - 6 = 0\) leads to \(x = 6\)

By solving these two simple equations, we find the potential solutions to the original quadratic equation. This step is straightforward and involves solving basic linear equations derived from the factors. Both values, 5 and 6, are possible values for \(x\) that satisfy the product equation equaling zero. Applying the zero-product property allows the determination of solutions in a clear and direct manner.

Solution Verification

Verification is a critical step in solving mathematical equations, as it confirms the solutions found are accurate by substituting them back into the original equation. For our quadratic equation exercise, we substitute the solutions \(x = 5\) and \(x = 6\) back into the original equation \(x^2 + 30 = 11 x\).

• First, substituting \(x = 5\) gives \(5^2 + 30 = 11 \times 5\), which simplifies to \(25 + 30 = 55\), confirming the equation holds true.
• Second, substituting \(x = 6\) yields \(6^2 + 30 = 11 \times 6\), simplifying to \(36 + 30 = 66\), again verifying the correctness.

Verification is essential because it ensures no computational errors occurred during the factoring or solving processes. This reflection step serves to validate the logical correctness of the entire solving procedure, ensuring confidence in the solutions derived.