Problem 68 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. $$\text { Solve: } u^{2}-u-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \( u = \frac{1 + \sqrt{5}}{2} \) or \( u = \frac{1 - \sqrt{5}}{2} \)

Step 1: Identify Coefficients

A quadratic equation is of the form \( ax^2 + bx + c = 0 \). In this equation, \( a = 1, b = -1, c = -1 \)

Step 2: Implement the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). Plug in the coefficients from step 1 into the formula: \( u = \frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)} \)

Step 3: Simplify the Equation

Solving the above equation gives: \( u = \frac{1 \pm \sqrt{5}}{2} \)

Key Concepts

Quadratic Formula

Understanding the quadratic formula is essential for solving quadratic equations, which are equations of the second degree, typically in the form of \( ax^2 + bx + c = 0 \). The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), provides the solution for the roots of a quadratic equation where 'a', 'b', and 'c' are coefficients. These coefficients determine the shape and position of the parabola when graphed.

To make the quadratic formula easy to work with, remember that 'b' will always be taken with a negative sign, \( \pm \) indicates that there will be two possible solutions, and the term under the square root is called the discriminant, which can tell us about the nature of the roots.

• If the discriminant is positive, there are two real and distinct roots.
• If it's zero, there is one real root.
• If it's negative, the roots are complex and not real.

When you're given a quadratic equation like \( u^2 - u - 1 = 0 \), you substitute the coefficients into the quadratic formula to find the values of 'u' that make the equation true.

Coefficients of a Quadratic Equation

Coefficients in a quadratic equation play a pivotal role in its properties and the solutions it yields. In the standard form \( ax^2 + bx + c = 0 \), 'a', 'b', and 'c' are referred to as the coefficients of the quadratic equation, with 'a' being the coefficient of the quadratic term \( x^2 \), 'b' the linear term \( x \), and 'c' the constant term.

Let us decode their meanings:

• The coefficient 'a' determines the opening direction and width of the parabola; if 'a' is positive, the parabola opens upwards, and if negative, it opens downwards.
• The coefficient 'b' affects the position of the parabola along the horizontal axis.
• The constant 'c' represents the y-intercept of the parabola, where it crosses the y-axis.

It's crucial to correctly identify these coefficients to apply the quadratic formula correctly. For the equation \( u^2 - u - 1 = 0 \), 'a' equals 1, 'b' is -1, and 'c' is also -1. This step may seem straightforward, but it is the foundation for effectively applying the quadratic formula and solving for the roots.

Simplifying Equations

Simplifying equations is a process that makes complex algebraic expressions easier to work with and understand. It's a critical step before solving them, and it involves combining like terms, reducing fractions, and clearing radicals if possible.

In the context of solving quadratic equations using the quadratic formula, simplifying involves calculating the discriminant, performing the addition or subtraction within the numerator, and dividing the result by the denominator meticulously to avoid any calculation errors.

• Combine like terms (if any) before applying the quadratic formula.
• Carry out operations inside the square root (called the discriminant) carefully.
• Look to simplify the resulting square root, if possible.
• Finally, divide by the denominator to obtain the simplified roots of the equation.

For instance, in the equation \( u = \frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)} \), simplification leads to \( u = \frac{1 \pm \sqrt{5}}{2} \), which are the solutions to the original quadratic equation. Remember, careful simplification can help avoid mistakes and reach the accurate solutions.