Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-24 x+35=0 $$

Step-by-Step Solution

Answer

The given equation represents a parabola.

Step 1: Identify the Conic Equation

The given equation is a quadratic equation in the form of \( ax^{2} + bx + c = 0 \). Our equation is \( 4x^{2} - 24x + 35 = 0 \). This equation is that of a parabola because it is a quadratic equation with no other variables involved (like \( y \)).

Step 2: Group and Factor Coefficient of x²

First, rewrite the equation as \( 4(x^{2} - 6x) + 35 = 0 \). This groups the quadratic and linear terms together for ease in completing the square.

Step 3: Complete the Square

To complete the square, take the coefficient of \( x \), which is \( -6 \), halve it to get \( -3 \), and then square it to ( 9 \ . Add and subtract \( 9 \) inside the parentheses: \[ 4(x^{2} - 6x + 9 - 9) + 35 = 0 \] Re-arrange to: \[ 4((x-3)^{2} - 9) + 35 = 0 \]

Step 4: Simplify the Equation

Distribute the \( 4 \) and simplify the equation: \[ 4(x-3)^2 - 36 + 35 = 0 \] becomes \[ 4(x-3)^2 - 1 = 0 \].

Step 5: Set Equation to Standard Form of a Parabola

Solve for \( 4(x-3)^2 \) by moving \( -1 \) to the other side to obtain \( 4(x-3)^2 = 1 \). This confirms the equation represents a parabola because it's a squared term equal to a constant.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards, downwards, or sideways, although in the situation of the equation given, it opens upwards or downwards. Parabolas are a specific type of conic section, which are curves obtained by slicing a cone with a plane.

• They can be represented by quadratic equations of the form \( ax^2 + bx + c = 0 \).
• The vertex of the parabola is its highest or lowest point, depending on its orientation.
• Parabolas can model numerous real-life situations such as the path of a projectile.

In the exercise, the given equation \(4x^2 - 24x + 35 = 0\) is identified as a parabola because it is a quadratic equation. Parabolas do not require additional variables (like \( y \)) to be classified as such when working in one dimension.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This process is extremely useful for solving quadratic equations, integrating functions, and analyzing the properties of parabolas. The basic steps in completing the square involve:

• Taking the coefficient of the linear term (\( x \)), halve it, and then square the result.
• Adding and subtracting this squared value inside the quadratic expression.
• Simplifying the expression to form a perfect square trinomial.

In our exercise, after grouping, the expression inside the parenthesis was \( x^2 - 6x \). The coefficient of \( x \) is \(-6\). Halving gives \(-3\) and squaring it results in \(9\). By adding and subtracting \(9\) within the expression, it becomes \((x-3)^2 - 9\). Completing the square helps to transform the equation to a standard and more manageable form.

Quadratic Equation

The quadratic equation is a foundational concept in algebra and conic sections. It represents a second-degree polynomial and has the general form:\[ ax^2 + bx + c = 0 \]Here are some key elements:

• \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\).
• It can be solved using different methods: factoring, completing the square, or using the quadratic formula.
• The solutions of a quadratic equation are called its roots, which can be real or complex numbers.

In the provided equation \(4x^2 - 24x + 35 = 0\), 'a' is 4, 'b' is -24, and 'c' is 35. Since this is a quadratic equation, it represents a parabola when no other variable is in the equation. It helps determine the shape's key characteristics, including its direction, vertex, and intercepts.