Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 4 x^{2}-x=1 $$

Step-by-Step Solution

Answer

The roots are irrational and unequal: \(x_1 = \frac{1 + \sqrt{17}}{8}\), \(x_2 = \frac{1 - \sqrt{17}}{8}\).

Step 1: Rewrite the Equation in Standard Form

The given quadratic equation is \(4x^2 - x = 1\). Subtract 1 from both sides to rewrite it in the standard form \(ax^2 + bx + c = 0\). This gives: \[4x^2 - x - 1 = 0\] where \(a = 4\), \(b = -1\), and \(c = -1\).

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[D = (-1)^2 - 4 \times 4 \times (-1) = 1 + 16 = 17\].

Step 3: Analyze the Discriminant

The discriminant \(D = 17\) is positive and not a perfect square. This implies that the roots of the quadratic equation are irrational and unequal.

Step 4: Finding the Real Roots

Since the discriminant is positive, the equation has real roots. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots. Substitute the values of \(a\), \(b\), and \(c\): \[x = \frac{-(-1) \pm \sqrt{17}}{2 \times 4} = \frac{1 \pm \sqrt{17}}{8}\].

Step 5: Provide the Real Roots

The roots of the equation are: \(x_1 = \frac{1 + \sqrt{17}}{8}\) and \(x_2 = \frac{1 - \sqrt{17}}{8}\). These are irrational and unequal.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. The general form of a quadratic equation is given by:

• \[ ax^2 + bx + c = 0 \]

where:

• \( a \), \( b \), and \( c \) are constants with \( a \) \( eq 0 \)

Since the highest degree term is quadratic, its graph is a parabola. Quadratic equations are fundamental to algebra and appear frequently in various mathematical problems.
They can have either two solutions, known as roots, or no real solutions. Understanding the characteristics of these roots is key to analyzing the behavior of the quadratic graph and solving the equation.

Real Roots

In the context of quadratic equations, real roots refer to the solutions that are real numbers. There are several scenarios for real roots based on the discriminant value:

• When \( D > 0 \) (positive), the quadratic equation has two distinct real roots. If \( D \) is not a perfect square, these roots are irrational.
• When \( D = 0 \), the equation has exactly one real root, which means the roots are real, rational, and equal.
• When \( D < 0 \), the quadratic equation has no real roots, indicating that the solutions are complex numbers.

Knowing whether the roots are real or complex helps predict the intersection point of the parabola with the x-axis.
Real roots imply the intersections are on the x-axis, providing a tangible visualization of the solutions.

Irrational Roots

Irrational roots occur when a quadratic equation has a positive discriminant that isn't a perfect square. This means the roots are real but cannot be expressed as exact fractions or integers—they include a square root term that doesn't simplify to a whole number.
For the equation \( 4x^2 - x - 1 = 0 \), the discriminant calculation \( D = 17 \) suggests irrational roots. Since 17 is not a perfect square, this indicates that the solutions for \( x \) involve \( \sqrt{17} \). This irrational feature adds complexity and depth to the analysis of quadratic equations.

Quadratic Formula

The quadratic formula is a tool for solving any quadratic equation. It is especially useful when factoring is difficult or impossible. The formula is:

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

Here, \( b^2 - 4ac \) is the discriminant, whose value determines the nature of the roots:

• Two distinct real roots if \( D > 0 \).
• One real repeated root if \( D = 0 \).
• No real roots if \( D < 0 \) (complex roots appear).

The formula not only provides the solutions but also gives insight based on the discriminant. This makes it a powerful method in solving quadratic equations. In our exercise, the formula reveals two irrational, unequal roots given the discriminant of 17. This formula is universally applicable, demonstrating its significance in algebra.