Problem 42 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$7 x^{2}+8 x+1=0$$

Step-by-Step Solution

Answer

The solutions to the equation are x = -0.14 and x = -1.0

Step 1: Identifying coefficients

From the given equation \( 7x^{2}+8x+1=0 \), we can identify the coefficients as: \( a=7, b=8, c=1 \).

Step 2: Calculate the Discriminant

The discriminant is calculated as \( D= b^{2}- 4ac \). Substituting our values in, we get \( D = 8^{2} - 4*7*1 = 64 - 28 = 36 \).

Step 3: Apply Quadratic Formula

We use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) We substitute a=7, b=8 and D=36 and get the solutions as \( x = \frac{-8 \pm \sqrt{36}}{2*7} \). Hence, the two solutions are \( x = \frac{-8 + 6}{14}= -0.14 \) and \( x = \frac{-8 - 6}{14}= -1.0 \)

Key Concepts

Discriminant

When dealing with quadratic equations, the discriminant is a crucial element. It helps us determine the nature of the roots of an equation.
For any quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula:

• \( D = b^2 - 4ac \)

In this calculation, a few scenarios can arise based on the value of \( D \):

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the roots are complex or imaginary.

Hence, in our given quadratic equation \( 7x^2 + 8x + 1 = 0 \), we calculated the discriminant to be 36. Since \( 36 > 0 \), this indicates the presence of two distinct real solutions. Evaluating this step not only gives us insight into the type of solutions we can expect but also guides us on how to proceed with solving the equation.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \). This ensures that the equation is indeed quadratic and has a parabolic graph.
Quadratic equations are prevalent in various fields, such as physics, engineering, and finance, due to their ability to model complex relationships.
For our specific problem, the equation \( 7x^2 + 8x + 1 = 0 \) represents a scenario where the coefficients \( a = 7 \), \( b = 8 \), and \( c = 1 \) are identified. Solving a quadratic equation involves finding the values of \( x \) that satisfy it. These values, or roots, depict the points where the parabola intersects the x-axis when graphed.
Utilizing methods such as factoring, completing the square, or the quadratic formula allows us to pinpoint these intersections. The quadratic formula, in particular, is versatile and works in all cases, making it an essential tool for tackling such equations.

Roots of Equation

The roots of a quadratic equation are the solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are where the parabola, represented by the quadratic equation, crosses the x-axis. Often, these are referred to as the x-intercepts.
The quadratic formula is a reliable method to find these roots, given by:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

Substituting the values of \( a \), \( b \), and \( D \) into the formula gives us the specific roots. For our example, substituting \( a = 7 \), \( b = 8 \), and \( D = 36 \) gives:

• \( x = \frac{-8 \pm \sqrt{36}}{14} \)

Resulting in the roots \( x = -0.14 \) and \( x = -1.0 \).
These values tell us that the parabola will intersect the x-axis at \( x = -0.14 \) and \( x = -1.0 \). Understanding and finding these roots provide insights into the solutions of quadratic equations, assisting in solving real-world problems where such equations occur.