Problem 78 - Algebra 2

Question

Evaluate \(b^{2}-4 a c\) for the given values of \(a, b,\) and \(c\). \(a=1, b=7, c=3\)

Step-by-Step Solution

Answer

The value of the expression is 37.

Step 1: Identify the Formula

The expression we need to evaluate is the discriminant of a quadratic equation, denoted as \(b^2 - 4ac\). This expression is often used to determine the nature of the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\).

Step 2: Substitute the Given Values

Insert the given values \(a = 1\), \(b = 7\), and \(c = 3\) into the formula \(b^2 - 4ac\):\[b^2 - 4ac = 7^2 - 4 \times 1 \times 3\]

Step 3: Calculate \(b^2\)

Calculate \(7^2\), which is the square of \(b\):\[7^2 = 49\]

Step 4: Calculate \(4ac\)

Calculate \(4 \times 1 \times 3\):\[4 \times 1 \times 3 = 12\]

Step 5: Subtract \(4ac\) from \(b^2\)

Subtract the result from Step 4 from the result in Step 3:\[49 - 12 = 37\]

Step 6: Conclude the Calculation

The final result of evaluating \(b^2 - 4ac\) when \(a=1\), \(b=7\), and \(c=3\) is 37.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means it involves the square of a variable. The standard form of a quadratic equation is given by:\[ ax^2 + bx + c = 0 \].Here, \( a, b, \) and \( c \) are coefficients, with \( a eq 0 \). It is crucial to have \( a eq 0 \) because if \( a = 0 \), we end up with a linear equation, not a quadratic.
The graphical representation of a quadratic equation is a parabola. The coefficient \( a \) determines the direction of the parabola:

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

Understanding quadratic equations is essential because they frequently arise in various fields such as physics, engineering, and economics.

Roots of Quadratic

A quadratic equation typically has two roots, which are the solutions to the equation \( ax^2 + bx + c = 0 \). These roots can be found using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
In this formula, the expression \( b^2 - 4ac \) is called the discriminant.The roots can be real or complex, and they can be identical or distinct:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there are two identical real roots, which means the roots are equal.
• If \( b^2 - 4ac < 0 \), the roots are complex numbers and no real solutions exist.

By calculating the roots, we can understand the solutions' nature and how they relate to the quadratic equation.

Nature of Roots

The nature of the roots of a quadratic equation is directly linked to the value of the discriminant \( b^2 - 4ac \). The discriminant gives us insight into the type of roots without needing to solve the equation completely.
Here's how the discriminant affects the roots:

• Positive Discriminant: When \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots. This indicates that the parabola intersects the x-axis at two points.
• Zero Discriminant: When \( b^2 - 4ac = 0 \), the quadratic equation has two identical real roots, also known as repeated roots. The parabola touches the x-axis at exactly one point.
• Negative Discriminant: When \( b^2 - 4ac < 0 \), the roots are complex and imaginary, meaning the parabola does not intersect the x-axis at any point.

This knowledge allows us to quickly assess the potential solutions and behavior of the equation, making it a powerful tool in algebra.