Problem 83 - Algebra and Trigonometry

Question

Solve equation by the method of your choice. $$ 2 x^{2}-x=1 $$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(2x^2 - x = 1\) are \(x1 = 1\) and \(x2 = -0.5\).

Step 1: Identifying the coefficients

Identify the coefficients for the terms in the quadratic equation, which are \(a = 2\), \(b = -1\) and \(c = -1\) in our case.

Step 2: Calculating the Discriminant

Calculate the discriminant, \(D = b^2 - 4ac\), which, in this case, would be \(D = (-1)^2 - 4 * 2 * (-1) = 1 + 8 = 9\).

Step 3: Using Quadratic Formula

If the Discriminant is greater than zero, the equation will have two distinct real roots. In this case, the roots can be found by plugging the values of \(a\), \(b\), and \(D\) in the Quadratic formula which is \(x = [-b ± sqrt(D)] / (2a)\). For this equation, the roots would be \(x = [1 ± sqrt(9)] / (4) = [1 ± 3] / 4\), which gives us \(x1 = 1\) when using the positive square root and \(x2 = -0.5\) when using the negative square root.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. It provides a straightforward method to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[x = \frac{-b \pm \sqrt{D}}{2a}\]where:

• \( a \), \( b \), and \( c \) are the coefficients of the equation
• \( D \) is the discriminant, calculated as \( b^2 - 4ac \)

The plus-minus (±) sign indicates the two potential roots of the equation. By substituting the values of \( a \), \( b \), and \( c \) from the quadratic equation into the formula, you will find the values of \( x \) that satisfy the equation. The quadratic formula is particularly useful when the quadratic equation cannot be easily factored.

Discriminant

The discriminant is a crucial part of the quadratic formula and helps us understand the nature of the roots of a quadratic equation. It is found inside the square root of the quadratic formula and is calculated as:\[D = b^2 - 4ac\]Here, \( b \), \( a \), and \( c \) are the coefficients from the equation \( ax^2 + bx + c = 0 \). The discriminant tells us the kind of roots we can expect:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is one real root, which means the roots are equal, or a double root.
• If \( D < 0 \), the roots are complex or imaginary, and not real roots.

By examining the discriminant, you can quickly determine how many roots an equation has and whether they are real or complex, aiding in understanding the solution without needing to calculate further.

Roots of an Equation

Roots of a quadratic equation are the values of \( x \) that make the equation true, essentially the solutions to the equation. For a typical quadratic equation like \( ax^2 + bx + c = 0 \), the roots can be found via different methods:

• Factoring: Simple if the equation is easily factorable.
• Quadratic Formula: Used when factoring is complex or impossible.
• Completing the Square: A method that involves rearranging the equation into a perfect square.

The roots can vary:

• Real Roots: When the discriminant is zero or positive. As in our example, with a discriminant of 9, we have real roots.
• Complex Roots: When the discriminant is negative, leading to complex numbers.

Finding the roots of an equation is vital as they often represent meaningful quantities in real-world problems, like time, speed, or other measurable factors.