Problem 12 - Algebra 1

Question

Using THE DISCRIMINANT Tell if the equation has two solutions, one solution, or no real solution. $$-\frac{1}{3} x^{2}+x+4=0$$

Step-by-Step Solution

Answer

The equation has two real solutions.

Step 1: Identify the coefficients a, b, and c

From the given equation \(-\frac{1}{3} x^{2}+x+4=0\), the coefficients are \(a = -\frac{1}{3}\), \(b = 1\), and \(c = 4\).

Step 2: Calculate the discriminant

Use the formula for the discriminant, which is \(b^2 - 4ac\). Substituting the values for a, b and c gives: \(1^2 - 4*(-1/3)*4 = 1 - (-16/3) = 1 + \frac{16}{3} = 1 + 5.33 = 6.33.\)

Step 3: Determine the number of solutions from the discriminant

Since the discriminant is positive (6.33), the given equation has two solutions.

Key Concepts

Quadratic Equation Solutions

When we talk about quadratic equation solutions, we are referring to the answers that satisfy the equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. A quadratic equation can have either two real solutions, one real solution, or no real solution. The nature of the solutions is determined by the value of the discriminant.

Solutions to a quadratic equation can be found graphically by plotting the quadratic function and identifying the points where it crosses the x-axis, which correspond to the roots of the equation. Algebraically, solutions can be obtained using the quadratic formula, factoring, or completing the square.

Calculating the Discriminant

Calculating the discriminant is an essential step in analyzing the nature of the solutions of a quadratic equation. The discriminant is given by the formula \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant can tell us:

• If \( D > 0 \) then the equation has two distinct real solutions.
• If \( D = 0 \) then the equation has exactly one real solution.
• If \( D < 0 \) then the equation has no real solutions; however, it has two complex solutions.

Learning to calculate the discriminant accurately is vital since it allows us to predict the number and type of solutions without explicitly solving the equation. It is a quick way to assess how many x-intercepts a parabola will have on a graph.

Quadratic Formula

The quadratic formula is a tool that provides the solutions to any quadratic equation. It is derived from the process of completing the square and is written as \( x = \frac{-b \pm \sqrt{D}}{2a} \), where \( D \) is the discriminant \( (b^2 - 4ac) \) and \( a \), \( b \), and \( c \) are the coefficients of the equation \( ax^2 + bx + c = 0 \). The symbol \( \pm \) means that there are two solutions:\