Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 28 x-49 x^{2}=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 0.2857 \)

Step 1: Identify coefficients a, b, and c

In our equation \( -49x^{2} + 28x - 4 = 0 \), a is -49, b is 28, and c is -4.

Step 2: Substitute the coefficients in the quadratic formula

Substitute a = -49, b = 28, and c = -4 in the formula \( x = \frac {-b ± \sqrt {b^2 – 4ac}} {2a} \). This gives us \( x = \frac {-28 ± \sqrt {(28)^2 – 4*(-49)*(-4)}} {2*(-49)} \).

Step 3: Simplify the equation

The equation simplifies to \( x = \frac {-28 ± \sqrt {784 – 784}} {-98} \) which further simplifies to \( x = \frac {-28} {-98} \).

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra. A quadratic equation is any equation that takes the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not equal to zero. The solutions to these equations are the values of \( x \) that make the equation true, known as the 'roots' of the equation.

To solve a quadratic equation, one may use the Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides a method to find the roots regardless of whether they are real or complex numbers. In the context of our exercise, by substituting the identified coefficients into the Formula, you arrive at an operable expression from which the roots can be calculated with further simplification.

Quadratic Equation Coefficients

Quadratic equation coefficients are the numerical factors of the terms in a quadratic equation. In the standard form \( ax^2 + bx + c = 0 \), the coefficient \( a \) is associated with the \( x^2 \) term, \( b \) with the \( x \) term, and \( c \) is the constant term. These coefficients dictate the shape and position of the quadratic graph when plotted.

• \( a \) affects the direction of the parabola (upward if positive, downward if negative) and its width (narrower for larger absolute values of \( a \)).
• \( b \) affects the position of the axis of symmetry and the location of the vertex of the parabola.
• \( c \) represents the y-intercept of the parabola.

Identifying these coefficients correctly is vital, as any mistake will lead to incorrect solutions. In our problem, once identified, we notice that the coefficient \( a \) is negative, which means our parabola opens downward, and this is reflective in the solutions.

Mathematical Simplification

Mathematical simplification involves reducing a complex expression into its simplest form, making it easier to understand or to further manipulate mathematically.

Working with the Quadratic Formula

Simplification in the context of the Quadratic Formula often involves several steps, such as combining like terms, factoring, and reducing fractions. This requires careful arithmetic to ensure that each step is performed accurately. In our example, after substituting the coefficients, simplification includes calculating the value within the square root and reducing the fraction \( \frac{-28}{-98} \) to its simplest form.

Simplification is not just about making expressions smaller or shorter; it's about clarity and making sure that the solution is presented in a form that is most useful for interpretation or further calculations. By simplifying, we obtained the final solutions, which in this case are real, rational numbers that can be used to understand the behavior of the quadratic function associated with the equation.