Problem 27 - Elementary Algebra

Question

For each problem below, evaluate \(b_{2}-4 a c\), given the following values for \(a, b\), and \(c\). $$ a=-3, b=-12, c=19 $$

Step-by-Step Solution

Answer

The value is 372.

Step 1: Understand the Expression

We need to evaluate the expression \(b^2 - 4ac\) using the given values for \(a\), \(b\), and \(c\). This expression is commonly part of the quadratic formula.

Step 2: Substitute the Values

Substitute the given values of \(a = -3\), \(b = -12\), and \(c = 19\) into the expression:\[b^2 - 4ac = (-12)^2 - 4(-3)(19)\]

Step 3: Evaluate \(b^2\)

Calculate \(b^2\) by squaring \(b\):\[(-12)^2 = 144\]

Step 4: Evaluate \(-4ac\)

Calculate \(-4ac\) using the substituted values:\[-4(-3)(19) = 4 \times 3 \times 19 = 228\]

Step 5: Calculate the Final Result

Add the values from Step 3 and Step 4:\[144 + 228 = 372\]

Step 6: Conclusion of Evaluation

The calculated value of the expression \(b^2 - 4ac\) is 372.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra used for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is expressed as:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]The discriminant, \(b^2 - 4ac\), plays a key role here. It's essential for determining the nature of the solutions to the quadratic equation. When evaluating a quadratic problem, the discriminant helps tell us about the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a double root).
• If \(b^2 - 4ac < 0\), the roots are complex.

Understanding how to use the quadratic formula effectively is crucial for solving such equations and interpreting the solutions.

Evaluate Expression

Evaluating expressions involves substituting given values into an algebraic equation to find a numerical result. In the context of quadratic equations, the expression \(b^2 - 4ac\) is a critical component of the quadratic formula. To evaluate it:1. Substitute the values of \(a\), \(b\), and \(c\) into the expression.2. Perform the arithmetic operations, such as squaring, multiplying, and subtracting, in a systematic manner.For example, if \(a = -3\), \(b = -12\), and \(c = 19\), the expression becomes:\[(-12)^2 - 4(-3)(19)\]Breaking down the steps ensures clarity and accuracy, allowing you to find the discriminant with ease.

Mathematics Problem Solving

Problem-solving in mathematics involves breaking down complex problems into manageable steps to find solutions. This structured approach is essential when dealing with quadratic equations, among other topics. Here's a typical process:

• Understand the problem and identify what's being asked.
• Recognize the form of equation or expression you are working with.
• Substitute known values methodically.
• Calculate carefully, taking note of operations like squaring and multiplication.
• Verify results against expected outcomes or characteristics.

By following a sequence, like evaluating \(b^2 - 4ac\), mathematical problem solving becomes systematic and less daunting. This ensures accuracy and develops greater comprehension of algebraic concepts.

Substitution in Algebra

Substitution is a fundamental technique in algebra, crucial for evaluating equations and expressions. It involves replacing variables with known numerical values to simplify or solve problems. Here's how substitution works:1. Identify the variables in the expression or equation.2. Replace these variables with the provided numerical values.3. Perform the necessary arithmetic operations.For example, substituting \(a = -3\), \(b = -12\), and \(c = 19\) into \(b^2 - 4ac\) transforms the expression into:\[(-12)^2 - 4(-3)(19)\]Simplifying these operations not only gives you the outcome, but also solidifies understanding of how expressions change from abstract to concrete values through substitution.