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Chapter 1: Problem 27

In exercises \(25-27,\) find all real roots of the given function. $$ y(x)=4 x^{2}-5 $$

Short Answer

Expert verified
The real roots are \( x = \frac{\sqrt{5}}{2} \) and \( x = -\frac{\sqrt{5}}{2} \).

Step by step solution

01

Write the Equation

The function is given as \( y(x) = 4x^2 - 5 \). To find the real roots, we set the function equal to zero: \( 4x^2 - 5 = 0 \).
02

Isolate the Squared Term

Add 5 to both sides of the equation to isolate the squared term: \( 4x^2 = 5 \).
03

Solve for the Squared Variable

Divide both sides by 4 to solve for \( x^2 \): \( x^2 = \frac{5}{4} \).
04

Find the Roots

Take the square root of both sides to find the values of \( x \). Remember to consider both the positive and negative roots: \( x = \pm\sqrt{\frac{5}{4}} \).
05

Simplify the Roots

Simplify the square root: \( x = \pm\frac{\sqrt{5}}{2} \).

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Key Concepts

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

Real Roots
Real roots of a quadratic equation are the values of the variable that make the equation equal to zero. They are the points where the graph of the quadratic function intersects the x-axis. For the quadratic \[ y(x) = 4x^2 - 5, \]to find its real roots, we set the equation to zero and solve for the variable \( x \). A real root indicates that the expression is satisfied at this value, with no imaginary components involved.
  • Real roots can be zero or more: A quadratic can have two, one, or no real roots. This depends largely on the discriminant (the part under the square root in the quadratic formula).
  • If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (also called a repeated or double root).
  • In this example, since the equation becomes \( x = \pm \frac{\sqrt{5}}{2} \), it has two real roots because \( \sqrt{5} \) is a real number.
Square Roots
The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). When solving quadratic equations, such as \( x^2 = \frac{5}{4} \), finding the square roots means determining the values of \( x \) that satisfy this equation.
When you take the square root of both sides while solving equations like in our exercise, it's important to consider both the positive and negative roots:
  • The square root of a positive number has both positive and negative alternatives due to the squaring property (e.g., \( (-3)^2 = 9 \) and \( 3^2 = 9 \)).
  • Therefore, when we calculate \( x = \pm \sqrt{\frac{5}{4}} \), we find two possible roots: \( x = \frac{\sqrt{5}}{2} \) and \( x = -\frac{\sqrt{5}}{2} \).
In mathematical problems involving quadratic equations, square roots are fundamental for simplifying expressions and finding potential solutions.
Quadratic Functions
Quadratic functions are polynomial functions of degree 2, generally in the form \( y = ax^2 + bx + c \). These functions graph as parabolas which open upwards if \( a > 0 \) or downwards if \( a < 0 \). The key properties of quadratic functions include finding roots, the vertex, and the axis of symmetry.
  • The **roots** are the x-values where the function intersects the x-axis. For a function in standard form, you find the roots by setting \( ax^2 + bx + c = 0 \). In our exercise, this aims at determining where \( 4x^2 - 5 = 0 \).
  • The **vertex** of a parabola represents the maximum or minimum point. For \( y = 4x^2 - 5 \), there is a minimum point as the parabola opens upwards.
  • The **axis of symmetry** is a vertical line that passes through the vertex, offering a mirror-like property across the parabola. For our function, the axis would be along \( x = 0 \) since there's no \( bx \) term shifting it left or right.
Understanding these properties explains how quadratic functions behave on a graph and offers insight into the equations that define them.

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