Problem 80 - College Algebra Essentials

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 solution for the given values \(a = 2\), \(b = 9\), and \(c = -5\) is -5.

Step 1: Evaluate the discriminant

The discriminant is the part of the formula under the square root, \(b^{2} - 4ac\). In this case, it can be computed as \(9^{2} - 4*2*(-5)\), which simplifies to 121.

Step 2: Evaluate the square root

The square root of the discriminant is \(\sqrt{121}\), which equals to 11.

Step 3: Final calculation

The complete expression can now be evaluated. Substitute values \(b = 9\), \(a = 2\), and the result of the square root into the equation to get \(-\frac{9 + 11}{2*2}\). This simplifies to \(-\frac{20}{4} = -5\).

Key Concepts

Discriminant

Understanding the discriminant is crucial when solving quadratic equations using the quadratic formula. The discriminant, represented by the symbol \( \Delta \), gives us important information about the nature of the solutions to a quadratic equation. It is the part under the square root in the quadratic formula \( - b \pm \sqrt{b^2 - 4ac} \), and is calculated as \(b^2 - 4ac\).

If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has exactly one real root, also known as a repeated or double root. However, if the discriminant is negative, the equation has two complex roots. Knowing this helps in predicting the solutions even before we carry out the actual calculation. For example, in our exercise, evaluating \( 9^2 - 4 \times 2 \times -5 \) results in a positive discriminant of 121, indicating that two real solutions exist for the quadratic equation.

Square Root

A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). Mathematically, this is represented as \( \sqrt{x} \). This operation is crucial when using the quadratic formula to solve quadratic equations. To compute a square root, you may either use a calculator or simplify by factoring if the number is a perfect square, like 121 in our original exercise. The square root of 121 is 11 because \( 11 \times 11 = 121 \).

Not all square roots are as straightforward to evaluate as perfect squares. If the number under the square root is not a perfect square, the result is an irrational number, and you would typically leave it in radical form or approximate it to a decimal value.

Evaluate Expressions

To evaluate an expression means to substitute the given numerical values for the variables and then perform the indicated operations. When evaluating the quadratic formula, you substitute the values for \( a \), \( b \), and \( c \), and then follow the operations step by step.

As seen in the original exercise solution, after finding the discriminant and calculating the square root, the final step is to place these values back into the formula and perform the arithmetic operations. By substituting \( a = 2 \), \( b = 9 \) and the square root of 121 (which is 11), into the quadratic formula, we carry out the division and subtraction to simplify the expression to \( -5 \), the solution to the quadratic equation.