Problem 63 - Intermediate Algebra

Question

Why must we change the equation \(3 x^{2}-2 x=4\) to \(3 x^{2}-2 x-4=0\) before applying the quadratic formula?

Step-by-Step Solution

Answer

The equation must equal zero for the quadratic formula to work.

Step 1: Understanding the Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation, which is of the form \( ax^2 + bx + c = 0 \). The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It is essential that the equation is set to equal zero to apply this formula.

Step 2: Identifying the Equation to Transform

The given equation is \( 3x^2 - 2x = 4 \). This is not in the form required for the quadratic formula because it is equated to 4, not 0.

Step 3: Transforming the Equation

Subtract 4 from both sides of the equation to set it equal to zero: \( 3x^2 - 2x - 4 = 0 \). This transformation aligns the equation with the standard quadratic form \( ax^2 + bx + c = 0 \).

Step 4: Finalizing the Quadratic Form

The transformed equation \( 3x^2 - 2x - 4 = 0 \) is now in the correct format for applying the quadratic formula, with \( a = 3 \), \( b = -2 \), and \( c = -4 \).

Key Concepts

Quadratic Formula

The quadratic formula is a steadfast tool in solving quadratic equations. It efficiently provides the solutions or roots for any quadratic equation that fits the model \( ax^2 + bx + c = 0 \). Its utility lies in its universality, applicable to every quadratic situation.The formula itself is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]When using the quadratic formula:

• Ensure the equation is in standard form where everything is set to equal zero
• Identify the coefficients: \(a\), \(b\), and \(c\)
• Proceed with substituting these values into the formula

This formula can look intimidating, but it systematically breaks down any quadratic obstacle once you get the hang of identifying the necessary components. Make sure to pay attention to signs and arithmetic details!

Equation Transformation

Equation transformation is a crucial step when preparing to use the quadratic formula. It involves rewriting the given equation to fit the standard quadratic form.Consider the equation \(3x^2 - 2x = 4\). The first step in transforming this equation is to subtract 4 from both sides, resulting in:\[3x^2 - 2x - 4 = 0\]By adjusting the equation to equal zero, you align it with the necessary format required by the quadratic formula. Here's why this transformation matters:

• It allows identification of the coefficients \(a\), \(b\), and \(c\)
• It ensures compatibility with the quadratic formula
• Makes the equation easier to manipulate mathematically

This step acts as a bridge between the initial equation and a successful application of the quadratic formula.

Standard Quadratic Form

The standard quadratic form is the notation \( ax^2 + bx + c = 0 \). This form is crucial because it lays the groundwork needed for utilizing the quadratic formula efficiently.Key aspects of the standard form include:

• The highest degree term should be \(x^2\)
• The equation simplifies the process of finding roots
• Coefficients \(a\), \(b\), and \(c\) are easily identifiable and usable in the quadratic formula

To translate any quadratic equation into this form, you often need to rearrange terms or move numbers from one side of the equation to the other, aiming always to isolate the quadratic polynomial on one side of the equation and set it to zero. By mastering the conversion to standard form, you unlock the full potential of the quadratic formula.