Problem 85 - Algebra A Combined Function

Question

Evaluate \(\sqrt{b^{2}-4 a c}\) for each set of values. $$ a=3, b=-1, c=-2 $$

Step-by-Step Solution

Answer

The value is 5.

Step 1: Evaluate Inside the Square Root

First, we substitute the values into the expression inside the square root. The expression is \(b^2 - 4ac\). Substitute \(a = 3\), \(b = -1\), and \(c = -2\) to get:\[b^2 - 4ac = (-1)^2 - 4(3)(-2)\]

Step 2: Calculate \(b^2\)

Now, we compute \(b^2\):\[(-1)^2 = 1\]

Step 3: Calculate \(-4ac\)

Next, calculate \(-4ac\):\[-4(3)(-2) = 24\]

Step 4: Combine Results Inside the Square Root

Add the results from steps 2 and 3:\[b^2 - 4ac = 1 + 24 = 25\]

Step 5: Compute the Square Root

Finally, take the square root of the result from step 4:\[\sqrt{25} = 5\]

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool in algebra for finding the roots of quadratic equations. A quadratic equation typically has the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The Quadratic Formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the values of \(x\) that satisfy the equation. It incorporates the discriminant, \(b^2 - 4ac\), which is the part of the formula inside the square root. The discriminant is key in determining the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root, or a repeated root.
• If \(b^2 - 4ac < 0\), the roots are complex and not real numbers.

Understanding the Quadratic Formula allows you to solve any quadratic equation quickly and efficiently. It can save time and reduce mistakes when dealing with complex coefficients.

Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because \(5 \times 5 = 25\). The square root is denoted by the symbol \(\sqrt{}\). It has important applications:

• Whenever you solve an equation and use the quadratic formula, you'll encounter the square root symbol.
• The square root helps us find values that satisfy quadratic expressions.
• In geometry, square roots are pivotal in determining distances and areas.

Calculating square roots can sometimes require estimation or a calculator, particularly when dealing with non-perfect squares, like \(\sqrt{2}\). In the exercise, the calculation of \(\sqrt{25}\) equals 5, since 25 is a perfect square.

Substitution Method

The Substitution Method is an essential algebraic tool for solving equations. It involves replacing variables with known values or expressions to simplify an equation. In the context of the problem, the method is used to evaluate expressions inside square roots:

• Step 1: Identify the expression where substitution will happen, like \(b^2 - 4ac\).
• Step 2: Replace the variables \(a\), \(b\), and \(c\) with the provided values. Here, \(a = 3\), \(b = -1\), and \(c = -2\).
• Step 3: Simplify the expression step-by-step to find the result.

Using substitution effectively helps in simplifying complex algebraic expressions and is fundamental for evaluating quadratics and other mathematical problems. It is a method that brings solutions closer by breaking down equations into manageable parts.

Quadratic Equation

A quadratic equation is any polynomial equation of degree two. Its standard form is \(ax^2 + bx + c = 0\), where \(a eq 0\). These equations graph as parabola shapes, opening either upwards or downwards:

• They have up to two solutions, which can be real or complex numbers.
• Quadratic equations model many real-world scenarios, such as projectile motion or optimizing areas and perimeters.
• The solutions can be found using various methods, such as factoring, completing the square, or the quadratic formula.

Understanding quadratic equations allows students to explore richer algebraic problems and solve them through intuitive or formulaic methods. Mastery of this concept opens doors to more advanced topics in mathematics.