Problem 43 - Basic Technical Mathematics with Calculus

Question

In finding the dimensions of a crate, the equation \(12 x^{2}-64 x+64=0\) is used. Solve for \(x,\) if \(x>2\).

Step-by-Step Solution

Answer

The value of \(x\) is 4, given that \(x > 2\).

Step 1: Write Down the Quadratic Equation

The quadratic equation given in the problem is \(12x^2 - 64x + 64 = 0\). We need to solve this equation to find the values of \(x\).

Step 2: Simplify the Quadratic Equation

Notice that all terms in \(12x^2 - 64x + 64 = 0\) can be divided by 4 to simplify: \(3x^2 - 16x + 16 = 0\). Use this simplified equation for further calculations.

Step 3: Identify Coefficients for the Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can be applied. Here, \(a = 3\), \(b = -16\), and \(c = 16\).

Step 4: Calculate the Discriminant

Compute the discriminant: \(b^2 - 4ac = (-16)^2 - 4 \times 3 \times 16\). This results in \(256 - 192 = 64\).

Step 5: Apply the Quadratic Formula

Substitute the values into the quadratic formula: \(x = \frac{-(-16) \pm \sqrt{64}}{2 \times 3}\). This simplifies to \(x = \frac{16 \pm 8}{6}\).

Step 6: Find the Roots

Calculate the two possible values of \(x\): \(x = \frac{16 + 8}{6} = \frac{24}{6} = 4\) and \(x = \frac{16 - 8}{6} = \frac{8}{6} = \frac{4}{3}\).

Step 7: Choose the Valid Root for the Problem Condition

Since we are looking for \(x > 2\), the valid solution is \(x = 4\).

Key Concepts

Understanding the Discriminant

When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), one of the key aspects to consider is the discriminant. The discriminant is found using the formula \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the quadratic equation. If the discriminant is positive, like in our example where it is 64, the quadratic equation has two distinct real roots.
When the discriminant equals zero, it means the equation has exactly one real root, often referred to as a repeated or double root. Lastly, if the discriminant is negative, the equation has no real roots and instead features two complex conjugate roots.
Understanding the discriminant helps predict how many solutions the quadratic equation will have and what type they will be. This knowledge is useful when assessing whether it's worth solving the equation by other means or considering alternative approaches like graphing.

The Quadratic Formula Demystified

The quadratic formula is a powerful tool for solving any quadratic equation. It is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The formula works for every quadratic equation, regardless of whether it can be factored simply or not. Here’s a quick breakdown of each part:

• The term \(-b\) adjusts the sign of the coefficient of \(x\).
• The symbol \(\pm\) indicates there are usually two possible solutions depending on whether you add or subtract.
• The calculated value of \(\sqrt{b^2 - 4ac}\) is crucial as it tells us about the discriminant, which we previously discussed.
• Finally, dividing by \(2a\) normalizes the solution based on the leading coefficient \(a\).

It's essential for students to remember that the quadratic formula is universal. That means it's applicable to all types of quadratic equations, including those that may seem more convoluted or hard to factor at first glance.

Solving Quadratic Equations Made Simple

A quadratic equation is any equation that can be expressed in the form \(ax^2 + bx + c = 0\). To solve it means to find the values of \(x\) that make the equation true. There are multiple methods for solving quadratic equations including factoring, completing the square, and using the quadratic formula.
Each method has its advantages depending on the specific equation at hand.

• Factoring is often the quickest method, but it requires the equation to be factorizable over the integers or real numbers.
• Completing the square offers a flexible technique that works universally, but it involves altering the equation into a perfect square trinomial, which might be complex for beginners.
• The quadratic formula is a go-to solution, especially when factoring is not straightforward, as it accommodates all scenarios.

The choice of method might depend on personal preference, the complexity of the coefficients, or the specific constraints of the problem, like whether we desire a particular solution range. In exercises such as the original one, after calculating using any method, always ensure to check which solution meets any additional condition given, such as finding values greater than a certain number.