Question

Use the discriminant to determine the number of real solutions of the quadratic equation. \(2 x^{2}-x-1=0\)

Short answer

The quadratic equation \(2x^{2}-x-1=0\) has two real solutions.

Step-by-step solution

Identify the coefficients

From the quadratic equation \(2x^{2}-x-1=0\), identify the coefficients as a=2, b=-1, and c=-1.

Calculate the discriminant

Plug these values into the discriminant formula, \(b^{2}-4ac\). This gives \((-1)^{2}-4*2*(-1) = 1 + 8 = 9\).

Interpret the discriminant value

Since the discriminant is 9, which is greater than 0, the quadratic equation has two distinct real solutions.

Key concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. This means it has the form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). The term \(ax^2\) gives it the degree of two, which is why it's called a quadratic equation. Quadratic equations are fundamental in many areas of mathematics, and you might see them in physics or economics as well.

In a quadratic equation, \(x\) represents the variable, and the highest power of \(x\) (which is 2) determines the shape of its graph, known as a "parabola." Understanding how to work with quadratic equations is an essential skill.

• The solutions to a quadratic equation are the values of \(x\) that make the equation true.
• These solutions can be found using various methods, including factoring, completing the square, or using the quadratic formula.

Real Solutions

The solutions to a quadratic equation can be real or complex numbers. When we talk about "real solutions," we mean solutions that are real numbers, which are numbers without any imaginary part.

The discriminant helps us determine the nature of these solutions. The discriminant is part of the quadratic formula, given by:\[ b^2 - 4ac \]

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If the discriminant is zero, it has exactly one real solution, also known as a repeated or double root.
• If the discriminant is negative, the equation has no real solutions, but rather two complex solutions.

In the given exercise, the discriminant is 9, which is positive. Therefore, there are two distinct real solutions. This knowledge is incredibly helpful in predicting the outcome without solving the equation completely.

Coefficients

Coefficients are numerical or constant factors in the terms of a polynomial, like a quadratic equation. In the formula \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are known as coefficients.

Understanding coefficients is important because they tell us how the different parts of the polynomial interact. For example in the provided quadratic equation \(2x^2 - x - 1 = 0\):

• \(a = 2\): This is the coefficient of \(x^2\), and it influences the direction and "width" of the parabola. If \(a\) were negative, the parabola would open downward.
• \(b = -1\): The coefficient of \(x\) affects the symmetry and the location of the "vertex" of the parabola on the graph.
• \(c = -1\): This is the constant term, and it shows where the parabola intersects the y-axis in a graph.

Each coefficient plays a critical role in defining the behavior and position of the quadratic graph. Adjusting them changes the shape and position of the parabola, affecting the real solutions we can find.