Problem 50 - College Algebra with Corequisite Support

Question

Show that the sum of the two solutions to the quadratic equation is \(-\frac{b}{a}\).

Step-by-Step Solution

Answer

The sum of the solutions to the quadratic equation is \(-\frac{b}{a}\).

Step 1: Recall the Standard Form

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable.

Step 2: Apply the Quadratic Formula

The solutions to the quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). These solutions are often denoted as \( x_1 \) and \( x_2 \).

Step 3: Find the Sum of the Solutions

The sum of the two solutions \( x_1 \) and \( x_2 \) can be calculated by adding them together: \[ x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a} \].

Step 4: Simplify the Expression

When you add the expressions for \( x_1 \) and \( x_2 \), the terms with the square root cancel each other out: \[ x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a} = \frac{-2b}{2a} = \frac{-b}{a} \].

Step 5: Conclude the Result

After simplification, it is shown that the sum of the solutions \( x_1 \) and \( x_2 \) to the quadratic equation is indeed \( -\frac{b}{a} \), which confirms the initial claim.

Key Concepts

Sum of Solutions

When dealing with quadratic equations, finding the sum of solutions can help you understand the properties of the equation's roots. If we consider a quadratic equation in the form of \( ax^2 + bx + c = 0 \), the sum of its solutions \( x_1 \) and \( x_2 \) can be found using simple algebraic manipulation.
One approach to find the sum of solutions is through Vieta's formulas, which states for a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of its roots are given by:

• The sum of the roots \( x_1 + x_2 = -\frac{b}{a} \)
• The product of roots \( x_1 \cdot x_2 = \frac{c}{a} \)

In this problem, we verified the sum of the solutions using the quadratic formula and algebraic simplification. By applying the Quadratic Formula and adding the roots, the terms involving the square root of the discriminant \( \sqrt{b^2 - 4ac} \) conveniently cancel out. This left us with the simplified expression \( \frac{-2b}{2a} \), which confirms that \( x_1 + x_2 = -\frac{b}{a} \). This result is consistent with Vieta's relationships.

Standard Form Quadratic Equation

Before diving into the sum of solutions, it's crucial to understand the standard form of a quadratic equation, which is expressed as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \), and \( x \) represents the variable.
This format is important because:

• It allows us to easily apply the quadratic formula to find the solutions.
• You can quickly identify the coefficients \( a \), \( b \), and \( c \), which are essential for various calculations like finding the vertex or axis of symmetry.

If an equation doesn't appear in the standard form, you must reorder and manipulate it to fit \( ax^2 + bx + c = 0 \).
This step ensures you can accurately apply mathematical formulas, including the quadratic formula, and effectively solve for \( x \). Proper understanding of this form helps in recognizing the importance of each part of the equation and how they interact to produce solutions.

Quadratic Equation Concepts

A quadratic equation is a polynomial equation of degree two, and these equations represent parabolas when graphed. Concepts connected to quadratic equations are foundational in algebra and explore how the equation behaves and can be solved.
Key aspects to understand include:

• Discriminant: Part of the quadratic formula \( b^2 - 4ac \) determines the nature of the roots – real vs. complex and distinct vs. repeated.
• Roots (solutions): Points where the parabola intersects the x-axis. These are found using the quadratic formula or factoring, when possible.
• Vertex form: Another way to express quadratics; revealing the parabola's maximum or minimum point.

Each quadratic equation presents a curve, and understanding its components is crucial. For instance, the discriminant not only helps in identifying the type of roots but also impacts the shape of the corresponding parabola.
Overall, mastering these concepts enables you to effectively tackle various algebraic problems involving quadratic equations, streamline the solution process, and understand deeper mathematical theories.