Problem 83 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ 7 x^{2}-8 x-6=0 $$

Step-by-Step Solution

Answer

The equation has two real solutions.

Step 1: Identify the coefficients

In our equation, \(7x^2 - 8x - 6 = 0\), we can see that \(a = 7\), \(b = -8\), and \(c = -6\)

Step 2: Calculate the Discriminant

The discriminant, \(D\), is given by \(D = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\) into this formula, we get \(D = (-8)^2 - 4*7*(-6)\). So, \(D = 64 + 168 = 232\)

Step 3: Interpret the Discriminant

Since \(D = 232\) which is greater than zero, we can conclude that there are two real solutions

Key Concepts

Understanding the Discriminant

The discriminant is a crucial part of solving quadratic equations. It is a value calculated from the coefficients of the equation, particularly in the quadratic formula. The formula for the discriminant is given by \( D = b^2 - 4ac \).
Here:

• \(b\) is the coefficient of the variable \(x\).
• \(a\) is the coefficient of \(x^2\).
• \(c\) is the constant term.

Calculating the discriminant helps us determine the nature of the solutions of a quadratic equation.
Depending on whether the discriminant is positive, zero, or negative, we can determine if there are two solutions, one solution, or no real solutions.

Exploring Real Solutions

Real solutions refer to the answers or roots of the quadratic equation that are real numbers. Whether a quadratic equation has real solutions can be understood by examining the discriminant.

• If \( D > 0 \): The equation has two distinct real solutions.
• If \( D = 0 \): The equation has exactly one real solution, also called a double root.
• If \( D < 0 \): The equation has no real solutions and instead has two complex or imaginary solutions.

Thus, in our example of \( 7x^2 - 8x - 6 = 0 \), the discriminant \( D = 232 \) indicates that there are two real solutions available.

Role of Coefficients in Quadratic Equations

Coefficients in a quadratic equation are the numbers that multiply the variable terms. They are essential in forming the equation and solving it.

• The quadratic equation is generally expressed as \( ax^2 + bx + c = 0 \).
• \(a\) is the coefficient of \(x^2\),\(b\) is the coefficient of \(x\), and \(c\) is the constant term.

Recognizing these coefficients allows us to plug them correctly into the discriminant formula.
This calculation reveals vital information about the solutions of the equation, guiding us towards finding the roots accurately.

The Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. It provides the exact values of \(x\) by plugging in the coefficients and the discriminant into one universal formula. The formula is:\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

• \(b\), \(a\), and \(c\) are the coefficients of the quadratic equation.
• The symbol \( \pm \) indicates that there are typically two possible values of \(x\).

By using this formula, one can easily calculate the roots of any quadratic equation, ensuring that even complex equations are accessible and solvable for students.