Problem 80 - Algebra and Trigonometry

Question

Exercises \(78-80\) will help you prepare for the material covered in the next section. Evaluate \(\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\) for \(a=2, b=9,\) and \(c=-5\)

Step-by-Step Solution

Answer

The final result is -5.

Step 1: Understanding the Formula

The provided formula \(\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\) is derived from the quadratic formula which is the solution of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. This formula is used to find the roots or zeros of the equation (i.e., those values of \(x\) for which the given equation becomes zero). Here, \(a = 2\), \(b = 9\), and \(c = -5\).

Step 2: Substitute the Values

Plug the given values of \(a\), \(b\), and \(c\) into the formula. So the expression becomes \(\frac{-9-\sqrt{(9)^{2}-4*(2)*(-5)}}{2*(2)}\).

Step 3: Simplify under the Square root

Next, simplify the terms under the square root. We have \((-9)^2 = 81\), and \(4*2*-5 = -40\). These results give us an updated expression, \(\frac{-9-\sqrt{81 - (-40)}}{2*(2)}\). Since subtracting a negative is the same as adding, the expression can be revised to \(\frac{-9-\sqrt{81 + 40}}{2*(2)}\).

Step 4: Further Simplification

Continue simplifying the equation. First, add 81 and 40 under the square root to get \(\frac{-9-\sqrt{121}}{2*(2)}\). Then, evaluate the square root of 121 to get 11, resulting in \(\frac{-9 - 11}{2 * 2}\).

Step 5: Final Calculation

Now, simplify the numerator by subtracting 11 from -9 to get -20. Then, simplify the denominator by multiplying 2 by 2 to get 4. This gives us \(\frac{-20}{4}\). Divide -20 by 4 to get the final result.

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental concept in algebra, and is expressed in a standard form:

• \[ ax^2 + bx + c = 0 \]

Here, \(a\), \(b\), and \(c\) are known as the coefficients of the equation. The term \(ax^2\) represents the quadratic term, \(bx\) the linear term, and \(c\) the constant term.
The presence of the squared term \(ax^2\) makes it a quadratic, distinguishing it from linear equations which do not contain squared terms. Solving a quadratic typically involves finding the values of \(x\) that satisfy the equation — these values are often referred to as the 'roots' of the equation.
Quadratic equations can be solved by different methods, but one of the most systematic and widely used methods is the quadratic formula. This formula provides an exact solution for the roots, given that the equation is set to zero.

Roots of the Equation

The roots of a quadratic equation are essentially the solutions to the equation, and these can be found using the quadratic formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our specific exercise, we are working with this formula to find the roots, but focusing on only one specific root expression:

• \[ \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]

The part inside the square root, known as the discriminant \((b^2 - 4ac)\), is crucial. It determines the nature of the roots. If this discriminant is:

• Positive: there are two distinct real roots.
• Zero: there is one real root (or a double root).
• Negative: the equation has two complex roots.

For our exercise, the process of solving involves substituting the values of \(a = 2\), \(b = 9\), and \(c = -5\) into the formula and calculating the expression step-by-step to uncover one of these roots.

Simplifying Expressions

Simplifying expressions is an essential step when working with complex algebraic equations like quadratic equations. It involves breaking down the expression into a more manageable form.
In the given exercise, we were required to simplify the expression resulting from the insertion of specific values into the quadratic formula.### Key Steps to Simplification:1. **Substitute Known Values:** Plug the given values of \(a=2\), \(b=9\), and \(c=-5\) into the base expression \(\frac{-b - \sqrt{b^{2}-4ac}}{2a}\). 2. **Simplify Under the Square Root:** Calculate \(b^2 = 81\) and \(4ac = -40\). Then add these to update the inside of the square root to \(81 + 40 = 121\). 3. **Evaluate the Square Root:** Solve for \(\sqrt{121} = 11\) simplifying the expression to \(\frac{-9 - 11}{4}\). 4. **Finalize Simplification:** Complete the arithmetic in the fraction to find \(\frac{-20}{4} = -5\).Throughout this process, simplification is vital as it leads us to the final, clean result that represents one of the roots of our quadratic equation. This step-by-step simplification ensures that errors are minimized and the algebraic expression is easily interpretable.