Problem 11 - Algebra 1

Question

Using THE DISCRIMINANT Tell if the equation has two solutions, one solution, or no real solution. $$-3 x^{2}+5 x-1=0$$

Step-by-Step Solution

Answer

The quadratic equation has two distinct real solutions.

Step 1: Identify the coefficients

The given quadratic equation is \(-3x^2 + 5x - 1 = 0\). Here, the coefficient 'a' of \(x^2\) is -3, 'b' the coefficient of 'x' is 5 and 'c', the constant term, is -1.

Step 2: Apply the formula for discriminant

Discriminant, D is given by \(D = b^2 - 4ac\). Substituting the values of a, b and c from the equation into the formula, we get \(D = (5)^2 - 4*(-3)*(-1)\).

Step 3: Calculate the discriminant

Simplifying the expression gives \(D = 25 - 12 = 13\).

Step 4: Interpret the result

Since the discriminant is greater than 0, the equation \(-3x^2 + 5x - 1 = 0\) has two distinct real solutions.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are polynomial equations of the second degree, typically taking the form \(ax^2 + bx + c = 0\). Here,

• \(a\), \(b\), and \(c\) are constants, known as coefficients.
• \(x\) is the variable.
• \(a\) is never zero because otherwise, the term \(ax^2\) would disappear, and the equation would no longer be quadratic.

Quadratic equations appear in various areas, such as physics, engineering, and economics, modeling situations from projectile motion to finding optimal solutions. Solving these equations involves finding the values of \(x\) that make the equation true, called the "roots" or "solutions" of the equation.

Identifying Real Solutions

Real solutions to quadratic equations are determined using the discriminant. The discriminant is a component of the quadratic formula. It helps to determine the nature of the roots of a quadratic equation. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The term \(b^2 - 4ac\) is known as the discriminant, often denoted by \(D\). The discriminant gives insights about the roots without having to solve the entire equation:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution (a repeated root).
• If \(D < 0\), there are no real solutions (solutions are complex).

Understanding the discriminant allows us to quickly assess how many and what type of solutions a quadratic equation will have.

Role of Coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are coefficients that substantially define the behavior of the equation:

• \(a\): The leading coefficient affects the parabola's width and direction. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
• \(b\): This coefficient influences the axis of symmetry of the parabola. The axis of symmetry can be calculated with \(-\frac{b}{2a}\).
• \(c\): The constant term affects the position of the parabola on the y-axis. It indicates where the parabola crosses the y-axis when \(x = 0\).

By examining these coefficients, you can predict some characteristics of the quadratic function's graph, such as its shape and position, which is a powerful tool in graph interpretation and application.