Problem 76 - Algebra A Combined Function

Question

For each quadratic equation, choose the correct substitution for \(a, b,\) and \(c\) in the standard form \(a x^{2}+b x+c=0 .\) \(x^{2}+5=-x\) a. \(a=1, b=5, c=-1\) b. \(a=1, b=-1, c=5\) c. \(a=1, b=5, c=1\) d. \(a=1, b=1, c=5\)

Step-by-Step Solution

Answer

The correct choice is d: \(a=1, b=1, c=5\).

Step 1: Rewrite the equation in standard form

Start by rearranging the given equation to match the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). The equation given is \(x^2 + 5 = -x\). Move all terms to one side to reformat it: \(x^2 + x + 5 = 0\).

Step 2: Identify coefficients a, b, and c

Once in standard form, identify the coefficients of \(x^2\), \(x\), and the constant term. In the equation \(x^2 + x + 5 = 0\), the coefficient of \(x^2\) is \(a = 1\), the coefficient of \(x\) is \(b = 1\), and the constant term is \(c = 5\).

Step 3: Choose the correct substitution

Now, compare the identified coefficients with the given options. Based on the values \(a = 1\), \(b = 1\), and \(c = 5\), the correct option is option d: \(a=1, b=1, c=5\).

Key Concepts

Standard Form of a Quadratic Equation

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). This is known as the standard form of a quadratic equation. In this form, \(a\), \(b\), and \(c\) are coefficients that determine the nature and position of the parabola representing the quadratic equation. Here, \(a\) is the coefficient of the \(x^2\) term, \(b\) is the coefficient of the \(x\) term, and \(c\) is the constant term. The standard form is essential for analyzing quadratic equations as it allows for easy identification of coefficients and facilitates solving the equation. By ensuring an equation is in standard form, we can use various methods to solve it, such as factoring, completing the square, or using the quadratic formula. Begin solving any quadratic equation by expressing it in this standard form.

Identifying Coefficients

Once the quadratic equation is in the standard form \(ax^2 + bx + c = 0\), identifying the coefficients \(a\), \(b\), and \(c\) becomes straightforward. These coefficients provide crucial information about the properties of the quadratic equation:

• \(a\): Governs the direction (upward or downward opening) and the width of the parabola.
• \(b\): Influences the position of the axis of symmetry of the parabola.
• \(c\): Represents the y-intercept of the quadratic graph.

For example, in the equation \(x^2 + x + 5 = 0\), it's easy to see that \(a = 1\), \(b = 1\), and \(c = 5\). Correctly identifying these coefficients is fundamental to both understanding the equation and solving it using various algebraic methods.

Rearranging Equations

Rearranging equations is a vital step in converting them into standard form. This involves moving all terms to one side of the equation so that one side equals zero. To rearrange the equation \(x^2 + 5 = -x\), you would move \(-x\) to the left side by adding \(x\) to both sides. This results in \(x^2 + x + 5 = 0\).

Steps to rearrange an equation:

• Combine like terms, making sure you keep variables and constants separate.
• Ensure all terms are on one side of the equation and set the opposite side to zero.
• Check that the equation follows the quadratic structure, \(ax^2 + bx + c = 0\).

Once rearranged, you can easily proceed with identifying coefficients or solving the quadratic equation. This step is critical, especially when equations are presented in forms that do not initially look like the standard quadratic form.

Solving Quadratic Equations

Solving quadratic equations means finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). There are several methods to achieve this, depending on the nature of the equation:

• Factoring: If the quadratic can be expressed as a product of its linear factors, set each factor to zero and solve for \(x\).
• Quadratic Formula: For any quadratic equation, the values of \(x\) can be found using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: This method involves rewriting the equation in a form that makes it a perfect square trinomial, which is then solvable by taking the square root of both sides.

Each method provides a pathway to solve the equation given its form and the coefficients \(a\), \(b\), and \(c\). Understanding these different techniques is fundamental to tackling various quadratic equations effectively.