Problem 21 - 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. $$ 4 x-1=4 x^{2} $$

Step-by-Step Solution

Answer

The roots are real, rational, and equal: \(x = \frac{1}{2}\).

Step 1: Write the Quadratic Equation in Standard Form

First, we need to rearrange the equation \(4x - 1 = 4x^2\) into the standard quadratic form, \(ax^2 + bx + c = 0\). This involves moving all terms to one side:\[4x^2 - 4x + 1 = 0\]

Step 2: Calculate the Discriminant

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

Step 3: Interpret the Discriminant

The value of the discriminant is \(0\). A discriminant of \(0\) indicates that the quadratic equation has real, rational, and equal roots. In this case, it falls under category (2): rational and equal roots.

Step 4: Find the Roots of the Equation

Since the discriminant is \(0\), the roots can be found using the formula \(x = \frac{-b \pm \sqrt{D}}{2a}\). We know \(D = 0\), so the formula simplifies to:\[x = \frac{-b}{2a}\]Substituting our values \(b = -4\) and \(a = 4\):\[x = \frac{-(-4)}{2 \cdot 4} = \frac{4}{8} = \frac{1}{2}\]This means the equation has one real, rational, and equal root: \(x = \frac{1}{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that you often encounter in algebra. Its general form is given by: \[ ax^2 + bx + c = 0 \] Here,

• \(a\), \(b\), and \(c\) are constants,
• \(a eq 0\) because if \(a = 0\), the equation would not be quadratic,
• \(x\) represents the variable or unknown we want to solve for.

The defining feature of a quadratic equation is that the variable \(x\) is squared. An equation becomes quadratic by the presence of this squared term. Quadratic equations can model various real-life scenarios, such as projectile motion or area calculations.

Real Roots

Real roots of a quadratic equation are the values of \(x\) that satisfy the equation by making it equal to zero. These roots can be found using several methods:

• Factoring: Express the quadratic as a product of two binomials, if possible.
• Quadratic Formula: Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square: Rewrite the equation to make it a perfect square trinomial.

The quadratic formula is particularly useful as it works for all quadratic equations, even when other methods fail. Whether or not the roots are real depends on the value of the discriminant \(D = b^2 - 4ac\). If \(D\) is non-negative, the roots are real.

Rational Roots

Rational roots are specific real roots that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\). Rational roots occur depending on the discriminant's value:

• If \(D = 0\), there is exactly one rational and repeated root given by \(x = \frac{-b}{2a}\).
• If \(D > 0\) and is a perfect square, the roots are rational and unequal.
• If \(D < 0\), there are no rational roots.

In the case of the equation \(4x^2 - 4x + 1 = 0\), the discriminant is \(0\), indicating a rational and repeated root, making the solutions simpler and straightforward to express.

Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This form is essential for a structured approach to solving and analyzing quadratic equations.Here's why the standard form is important:

• Identify Coefficients: It makes it easy to identify \(a\), \(b\), and \(c\) for use in the quadratic formula.
• Discriminant Calculation: It aligns with the discriminant formula \(D = b^2 - 4ac\), necessary for determining the nature of roots.
• Simplification: Equation manipulation tasks become systematic, aiding in employing consistent strategies like factoring or completing the square.

When a quadratic equation isn't in standard form, we need to rearrange it, as shown in the exercise where \(4x - 1 = 4x^2\) is reorganized into \(4x^2 - 4x + 1 = 0\). Being in this form helps us employ various methods to solve the equation effectively.