Problem 77 - Algebra and Trigonometry

Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. $$ 2 x^{2}-11 x+3=0 $$

Step-by-Step Solution

Answer

The discriminant is 97. The equation has two real solutions.

Step 1: Identify the coefficients a, b, c

The given equation is in the form \( ax^2 + bx + c = 0 \). So, \( a = 2, b = -11, and c = 3 \).

Step 2: Calculate the discriminant

The formula for the discriminant is \( D = b^2 - 4ac \). Substituting \( a, b, c \) gives \( D = (-11)^2 - 4*2*3 = 121 - 24 = 97 \).

Step 3: Determine number and type of solutions

Since \( D > 0 \), the equation has two real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest exponent of the variable, usually denoted as \( x \), is 2. The standard form of a quadratic equation is:
\[ ax^2 + bx + c = 0 \]where:

• \( a \), \( b \), and \( c \) are constants called coefficients.
• \( x \) represents the variable that you solve for.
• \( a eq 0 \) is crucial because if \( a \) is zero, the equation becomes linear, not quadratic.

Quadratic equations appear in many real-world contexts, such as in physics for motion problems, or in finance for certain types of profit maximization tasks. Solving quadratic equations can be done by factorizing, using the quadratic formula, or by completing the square. Understanding the general structure and components of a quadratic equation is fundamental for solving them accurately.

Real Solutions

In the context of quadratic equations, solutions refer to the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These solutions are also known as the roots of the equation. There are a few scenarios regarding the nature of these solutions based on the discriminant, \( D \), which is calculated as:
\[ D = b^2 - 4ac \]The nature of the solutions is determined as follows:

• If \( D > 0 \), the equation has two distinct real solutions. This indicates the graph of the quadratic function crosses the \( x \)-axis at two different points.
• If \( D = 0 \), there is exactly one real solution, also called a repeated or double root. This means that the graph touches the \( x \)-axis and "bounces" back, indicating the vertex of the parabola lies on the \( x \)-axis.
• If \( D < 0 \), there are no real solutions; instead, the solutions are complex or imaginary, which means the graph does not intersect the \( x \)-axis at all.

Thus, the discriminant is a critical component in predicting how many real solutions a quadratic equation will have and in understanding the graphical representation of the equation.

Coefficients in Algebra

Coefficients are numerical or constant terms in algebra that multiply the variable terms. In the quadratic equation form \( ax^2 + bx + c = 0 \), the coefficients are:

• \( a \), which is the coefficient of \( x^2 \)
• \( b \), which is the coefficient of \( x \)
• \( c \), which is the constant term without a variable

The coefficients play a crucial role in determining the shape and position of the parabola when the quadratic equation is graphed. Specifically:
- \( a \) affects the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. If negative, it opens downwards.
- \( b \) influences the location of the parabola's axis of symmetry.
- \( c \) denotes the \( y \)-intercept of the graph, which is the point where the parabola crosses the \( y \)-axis.Understanding coefficients helps in both solving equations algebraically and interpreting the graphical features of their representations.