Problem 40 - Algebra 1

Question

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$a=-2, b=8, c=-8$$

Step-by-Step Solution

Answer

The solution is 0.

Step 1: Substitution

Substitute the given values of \(a\), \(b\), and \(c\) into the formula: \(\sqrt{b^{2}-4 a c} = \sqrt{8^{2}-4(-2)(-8)}\).

Step 2: Simplification

Reduce the formula inside the square root by simplifying the multiplication and subtraction operations: \(\sqrt{64 - 64}\).

Step 3: Square root computation

Compute the square root of the result from Step 2: \(\sqrt{0}\).

Key Concepts

Substitution

In solving algebraic problems, substitution is a critical method where you replace variables with their specific values. This technique is especially useful when working with formulas and equations. For example, when evaluating the expression \(\sqrt{b^{2}-4ac}\) with the values given for \(a\), \(b\), and \(c\), you directly insert these numerical values into the respective variable positions. Here's how it's done:

Start by laying out the original expression: \(\sqrt{b^{2}-4ac}\). Then systematically replace each variable as follows:

• Replace \(b\) with 8 to get \(8^{2}\)
• Substitute \(a\) with -2 and \(c\) with -8, which you apply to the \(4ac\) part to get \(4(-2)(-8)\).

As a result, you'd have an expression that reads \(\sqrt{8^{2}-4(-2)(-8)}\), where all variables have been substituted. Always take care to maintain the correct order of operations and signs while substituting to avoid errors.

Simplification

Simplification involves reducing expressions into their simplest form, making them easier to work with or solve. After substituting variables with their values, the expression often requires further simplification. This means performing arithmetic operations like multiplication, division, addition, or subtraction that are part of the expression.

In our case, after substitution, simplify \(\sqrt{8^{2}-4(-2)(-8)}\) as follows:

• Calculate \(8^{2}\), which is 64.
• Multiply the numbers inside the parentheses: \(4(-2)(-8)\) equals to 64 as well since \(4\times -2\times -8 = 64\).

Therefore, the expression becomes \(\sqrt{64 - 64}\). This indicates that simplification is not always about mechanical calculations but also about seeing the structure of the expressions to find that, in this case, 64 minus 64 is simply 0. Now the expression is dramatically simpler as \(\sqrt{0}\), which is more straightforward to evaluate.

Quadratic Formula

The quadratic formula \(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) is a powerful tool used to find the roots of any quadratic equation, expressed as \(ax^{2} + bx + c = 0\). However, the piece \(\sqrt{b^{2}-4ac}\), known as the discriminant, is crucial on its own as it determines the nature of the roots. Our exercise involves evaluating this discriminant.

After simplification, we found that \(\sqrt{b^{2}-4ac}\) equals \(\sqrt{0}\) for the given values of \(a\), \(b\), and \(c\). In the context of the quadratic formula, a discriminant of 0 implies that there is exactly one real root for the given quadratic equation, or in other words, the roots are real and equal. It's essential to understand that the discriminant provides valuable insights into the quadratic equation without solving it completely. For the sake of our exercise, we focused on evaluating the discriminant part only, but the quadratic formula can be applied to find the actual roots if the full quadratic equation were given.