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Chapter 8: Problem 7

Evaluate each expression. a. \(\frac{-2 \pm \sqrt{2^{2}-4(1)(-8)}}{2(1)}\) b. \(\frac{-(-1) \pm \sqrt{(-1)^{2}-4(2)(-4)}}{2(2)}\)

Short Answer

Expert verified
Solutions for a are 2 and -4; solutions for b are \(\frac{1 + \sqrt{33}}{4}\) and \(\frac{1 - \sqrt{33}}{4}\).

Step by step solution

01

Identify the Expressions

We have two separate expressions to evaluate:a. \(\frac{-2 \pm \sqrt{2^{2}-4(1)(-8)}}{2(1)}\)b. \(\frac{-(-1) \pm \sqrt{(-1)^{2}-4(2)(-4)}}{2(2)}\)
02

Simplify Expression a

For expression a:- Identify each component: \(-b = -2\), \(b = 2\), \(a = 1\), and \(c = -8\).- Calculate the discriminant: \(b^2 - 4ac = 2^2 - 4(1)(-8)\).- Simplification gives: \(4 + 32 = 36\).- Calculate the square root: \(\sqrt{36} = 6\).- Apply the values into the quadratic formula: \(\frac{-2 \pm 6}{2}\).
03

Evaluate Both Solutions for Expression a

Now, perform the calculations:- Positive case: \(\frac{-2 + 6}{2} = \frac{4}{2} = 2\).- Negative case: \(\frac{-2 - 6}{2} = \frac{-8}{2} = -4\).Thus, the solutions for a are 2 and -4.
04

Simplify Expression b

For expression b:- Identify each component: \(-b = -(-1) = 1\), \(b = -1\), \(a = 2\), and \(c = -4\).- Calculate the discriminant: \((-1)^2 - 4(2)(-4) = 1 + 32 = 33\).- Calculate the square root: \(\sqrt{33}\).- Apply into the quadratic formula: \(\frac{1 \pm \sqrt{33}}{4}\).
05

Evaluate Both Solutions for Expression b

Evaluate the solutions:- Positive case: \(\frac{1 + \sqrt{33}}{4}\).- Negative case: \(\frac{1 - \sqrt{33}}{4}\).These are the solutions for expression b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
The discriminant is a key part of the quadratic formula, which determines the nature of the roots of a quadratic equation. In a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by the formula \( b^2 - 4ac \). This value tells us a lot about the solutions of the equation because it helps us understand the nature of the roots:
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
  • If the discriminant is negative, there are no real roots, but instead two complex conjugate roots.
Understanding the discriminant is crucial because it allows us to predict whether we will have real, distinct, or complex roots without solving the entire equation.
Roots of Quadratic Equations
Finding the roots of a quadratic equation is one of the main goals when solving these equations. The roots are the solutions to the equation \( ax^2 + bx + c = 0 \) and can be found using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]By substituting the coefficients \( a \), \( b \), and \( c \) from the quadratic equation into this formula, we get the values of \( x \) that satisfy the equation. These roots can be:
  • Real and distinct if the discriminant is positive, which means the graph of the quadratic function intersects the x-axis at two points.
  • Real and repeated if the discriminant is zero, in which case the vertex of the parabola touches the x-axis.
  • Complex and not real if the discriminant is negative, indicating the parabola does not intersect the x-axis at all.
Developing a strong grasp of this topic aids in solving quadratic equations effectively and understanding the nature of their graphs.
Square Roots
Square roots are a fundamental concept in mathematics, especially when solving quadratic equations. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, \( \sqrt{36} = 6 \) because \( 6 \times 6 = 36 \).
  • Square roots appear in the quadratic formula under the square root sign, \( \pm \sqrt{b^2 - 4ac} \).
  • The positive and negative symbols (\( \pm \)) represent two possible solutions: one with the positive square root and another with the negative.
  • The square root of the discriminant influences the nature of the roots, as we discussed earlier.
The ability to calculate square roots accurately is essential, especially when the discriminant is a perfect square, simplifying the process of finding exact numerical roots. Mastery of square roots supports a deeper understanding of quadratic solutions and strengthens overall math abilities.

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