Problem 84 - Algebra and Trigonometry

Question

Solve equation by the method of your choice. $$ 3 x^{2}-4 x=4 $$

Step-by-Step Solution

Answer

The solutions of the equation \(3x^{2}-4x-4=0\) are \(x=2\) and \(x=-\frac{2}{3}\).

Step 1: Rewrite Equation as Standard Form

Start by re-writing the standard form \(ax^{2}+bx+c=0\), so \[3x^{2}-4x-4 = 0\] this will make applying the quadratic formula easier.

Step 2: Apply Quadratic Formula

Once you have the equation in standard form with \(a=3\), \(b=-4\), and \(c=-4\), you can apply the quadratic formula \(\frac{-b±\sqrt{b^{2}-4ac}}{2a}\), which will solve for \(x\). The formula will look like this: \[x=\frac{-(-4) \pm \sqrt{(-4)^{2}-4*3*(-4)}}{2*3}\]

Step 3: Simplify Inside of the Square Root

Before proceeding with solving the equation, simplify the calculations inside the square root: \(16+48=64\). Then replace it back into the equation \[x=\frac{4±\sqrt{64}}{6}\].

Step 4: Solve for X

After you've simplified the square root, perform the final calculations, which gives you two potential answers for \(x\), using plus and minus operators: \[x=\frac{4+8}{6}=2\] and \[x=\frac{4-8}{6}=-\frac{2}{3}\]

Key Concepts

Understanding Quadratic Equations

A quadratic equation is an essential concept in algebra. It's a type of polynomial equation distinguished by the highest power of its variable being 2. This gives the equation a characteristic curve shape known as a parabola when graphed. A basic quadratic equation is typically written as:

• \(ax^2 + bx + c = 0\)

where:

• \(a\): coefficient of \(x^2\), it cannot be zero.
• \(b\): coefficient of \(x\).
• \(c\): constant term.

Quadratic equations represent a wide range of phenomena, from projectile motion in physics to financial calculations in business. Understanding how to solve them is a fundamental skill.

Transforming to Standard Form

To effectively solve a quadratic equation, it's helpful to rewrite it in its standard form: \(ax^2 + bx + c = 0\). This rearrangement creates a clearer path for applying mathematical methods, such as completing the square or using the quadratic formula. Let's take an example:Given the equation: \(3x^2 - 4x = 4\),First, move all terms to one side of the equation to set it to zero:

• Subtract 4 from both sides: \(3x^2 - 4x - 4 = 0\)

Now, it is in standard form, making it ready for applying various solution techniques. Working with equations in this format ensures consistency and correctness in further calculations.

Solving Equations Using the Quadratic Formula

The quadratic formula provides a straightforward method to find solutions to any quadratic equation. When the equation is in standard form \(ax^2 + bx + c = 0\), the quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula involves several important steps:

• Calculate the values of \(b^2 - 4ac\), known as the discriminant.
• The discriminant helps determine the nature of the roots.
• Solve for \(x\) by simplifying the expression.

Let's solve the example equation \(3x^2 - 4x - 4 = 0\):

• Here, \(a = 3\), \(b = -4\), and \(c = -4\).
• First compute the discriminant: \((-4)^2 - 4 \cdot 3 \cdot (-4) = 64\).
• Next, substitute these into the formula: \(x = \frac{4 \pm \sqrt{64}}{6}\).
• This results in the solutions \(x = 2\) and \(x = -\frac{2}{3}\).

By following these steps, you can conclude that quadratic equations can be resolved effectively using the quadratic formula.