Problem 3 - Algebra 1

Question

Find the discriminant for the equation. Then tell if the equation has two solutions, one solution, or no real solution. $$3 x^{2}-2 x+5=0$$

Step-by-Step Solution

Answer

The discriminant is -56, so the equation has no real solutions.

Step 1: Identify the coefficients a, b, c

The quadratic equation given is \(3x^2 - 2x + 5 = 0\). From this, the coefficients can be identified as \(a = 3\), \(b = -2\), and \(c = 5\). These values will be substituted into the formula for the discriminant.

Step 2: Calculate the discriminant 'D'

The formula for the discriminant 'D' is \(D = b^2 - 4ac\). Substituting the identified coefficients, we find \(D = (-2)^2 - 4*3*5 = 4 - 60 = -56\).

Step 3: Determine the number of solutions

The value of the discriminant, 'D', is used to determine the number of solutions for the equation. If 'D' is greater than 0, there are 2 distinct real solutions. If 'D' is equal to 0, there is exactly one real solution. If 'D' is less than 0, which is true in this case (since D=-56), there are no real solutions. This means the given equation has no real solutions.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. The formula is derived from the process of completing the square and can solve any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is given by:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
This formula calculates the roots of the equation by taking into account the coefficients 'a', 'b', and 'c'. The symbol \(\pm\) indicates that the equation may have two solutions, which correspond to the addition and subtraction of the square root term.

Number of Solutions

In any quadratic equation, the number of real solutions can be determined by the discriminant, which is the expression under the square root in the quadratic formula: \(D = b^2 - 4ac\).

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

Understanding the discriminant is critical as it provides insight into the nature of the solutions without actually solving the equation. It's a quick way to ascertain whether the roots are real or complex, and if real, whether they are distinct or coincident.

Coefficients

Coefficients are the numerical factors of the terms in a polynomial equation. In a quadratic equation of the form \(ax^2 + bx + c = 0\), 'a' is the coefficient of \(x^2\), 'b' is the coefficient of 'x', and 'c' is the constant term.

• The coefficient 'a' determines the parabola's direction (upward for positive 'a', downward for negative 'a') and how 'steep' or 'wide' the parabola will be.
• The coefficient 'b' impacts the location of the vertex of the parabola along the x-axis.
• The constant term 'c' indicates the y-intercept of the graph of the quadratic function.

The values of these coefficients play a pivotal role in defining the shape and the position of the parabola on the graph. They are also integral in calculating the discriminant, which as explained previously, is used to determine the nature of the solutions to the equation.