Problem 66 - Algebra 1: Concepts and Skills

Question

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$ a=2, b=4, c=-6 $$

Step-by-Step Solution

Answer

The evaluated result of \( \sqrt{b^{2}-4 a c} \) for a=2, b=4, and c=-6 is 8.

Step 1: Identify the values of a, b and c

From the question, it's clear that we have a=2, b=4, and c=-6.

Step 2: Substitute the values into the expression

The expression can be rewritten with these values as \( \sqrt{4^{2}-4*2*(-6)} \).

Step 3: Calculate the square and multiplication

Calculate the square of 4 and the multiplication of 4, 2 and -6: \( \sqrt{16-(-48)} \). Since the two numbers are being subtracted and the second number is negative, it can be rewritten as \( \sqrt{16+48} \).

Step 4: Perform the addition inside the square root

Perform the addition inside the square root to get \( \sqrt{64} \).

Step 5: Calculate the Square Root

The square root of 64 is 8.

Key Concepts

Discriminant

The discriminant is a key component in the quadratic formula that helps determine the nature of the roots of a quadratic equation. It's found in the formula for the quadratic equation as the part under the square root sign: \( b^2 - 4ac \). The discriminant can reveal whether a quadratic equation has two real roots, one real root (also known as a repeated root), or two complex roots.

• When the discriminant is positive, the equation has two distinct real roots.
• If it equals zero, there is exactly one real root, and it is repeated.
• When it's negative, the equation has two complex roots.

In this exercise, the discriminant is calculated as \( 4^2 - 4 \times 2 \times (-6) \). This simplifies to \( 16 + 48 \), resulting in a positive 64, clearly indicating our quadratic equation has two distinct real roots.

Square Root

The square root is a mathematical function that finds a number which, when multiplied by itself, gives the original number. In the context of quadratic equations and their solutions, the square root is part of the quadratic formula and allows us to solve for the roots after computing the discriminant.

In the exercise, after calculating the discriminant as 64, we have to find \( \sqrt{64} \). Finding the square root of 64 involves looking for a number that, when squared, gives us 64, which is 8. The process of solving \( \sqrt{64} \) involves recognizing perfect squares or utilizing a calculator for non-perfect squares.

Quadratic Equation

A quadratic equation is a polynomial of degree two. These can take the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown. Solving a quadratic equation often involves finding the values of \( x \) that satisfy the equation, which can be done using several methods like factoring, completing the square, or the quadratic formula.

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is often the go-to solution method when the equation doesn't easily factor. It involves calculating the discriminant and taking its square root, as we've discussed.

• The plus-minus symbol indicates that there are generally two solutions.
• Being familiar with how to manipulate square roots and work with the discriminant simplifies solving these equations.

In our exercise, though not directly solving a quadratic equation, we evaluated a key part of it which is crucial for understanding quadratic equations more deeply.