Problem 80 - Algebra 2

Question

Evaluate \(b^{2}-4 a c\) for the given values of \(a, b,\) and \(c\). \(a=2, b=-9, c=-5\)

Step-by-Step Solution

Answer

The value of the expression is 121.

Step 1: Substitute Given Values into the Expression

To evaluate the expression \( b^2 - 4ac \), first substitute the given values for \( a, b, \) and \( c \). We have \( a = 2 \), \( b = -9 \), and \( c = -5 \). Substitute these into the equation: \[ b^2 - 4ac = (-9)^2 - 4 \times 2 \times (-5) \]

Step 2: Calculate \(-9^2\)

Calculate the square of \( b \), which is \( -9 \) in this case. \[ (-9)^2 = 81 \]

Step 3: Calculate \(4 \times 2 \times (-5)\)

Next, compute the product \( 4 \times a \times c \), where \( a = 2 \) and \( c = -5 \). The calculation is as follows: \[ 4 \times 2 \times (-5) = -40 \]

Step 4: Simplify the Expression

Now, substitute the results from Step 2 and Step 3 back into the expression: \[ 81 - (-40) \]. When subtracting a negative number, it becomes positive, so this simplifies to \( 81 + 40 \).

Step 5: Evaluate the Final Expression

Add the numbers from the simplified expression in Step 4. \[ 81 + 40 = 121 \]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial that is characterized by the highest exponent being a square, or 2. In its standard form, it is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This ensures that the equation is indeed quadratic and not linear.

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

The quadratic equation is essential in mathematical modeling because it describes the path of parabolic trajectories. It appears frequently in physics, economics, and geometry. When solving a quadratic equation, one of the main goals is to find the values of \( x \) that make the equation true, commonly known as the solutions or roots.
The quadratic formula, derived from completing the square of the general quadratic equation, is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). This formula demonstrates the importance of the discriminant \( b^2 - 4ac \), as it determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), two distinct real roots exist.
• If \( b^2 - 4ac = 0 \), there is one real repeated root.
• If \( b^2 - 4ac < 0 \), the equation has two complex roots.

Parabolic Graph

A parabolic graph represents a quadratic equation visually. Its distinctive U-shaped curve is called a parabola. The basic structure of a parabola is influenced by the coefficients \( a \), \( b \), and \( c \) in the quadratic equation \( ax^2 + bx + c = 0 \).
The "opening" direction of the parabola (upward or downward) is determined by the sign of the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

The vertex of the parabola, which is its highest or lowest point, provides important information about the function's maximum or minimum value. It can be found using the formula \( x = -\frac{b}{2a} \), and substituting this back into the quadratic equation to find the corresponding \( y \)-value.
Graphing the parabola helps visually interpret the effects of different values of \( a \), \( b \), and \( c \), and quickly shows the number of roots and their approximate values when the graph intersects the x-axis.

Coefficients of a Polynomial

The coefficients of a polynomial are the numerical factors that multiply the variables in the equation. In a quadratic polynomial of the form \( ax^2 + bx + c \), each letter represents a coefficient:

• \( a \) is the coefficient of the squared term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term, or the coefficient of \( x^0 \).

These coefficients influence the behavior and graph of the polynomial. They play a critical role in determining the shape and position of the parabola. For instance:

• The absolute value of \( a \) affects the width of the parabola; larger absolute values result in a narrower graph, while smaller ones give a wider graph.
• The coefficient \( b \) impacts the symmetry and slope of the parabola.
• The constant \( c \) affects the vertical placement of the parabola on the graph.

Understanding these coefficients allows you to predict the general behavior of the quadratic function without fully solving the equation. This can be very helpful in applications across various fields such as physics, engineering, and finance, where quick approximations may be needed.