Problem 68 - Algebra and Trigonometry

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 roots of the equation \(u^{2}-u-1=0\) are \(u = \frac{1 + \sqrt{5}}{2}\) and \(u = \frac{1 - \sqrt{5}}{2}\).

Step 1: Understand the quadratic formula

The quadratic formula is given by \[ u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where a, b, c are coefficients of the quadratic equation \( ax^{2} + bx + c = 0 \). Here \( a = 1 \), \( b = -1 \), and \( c = -1 \).

Step 2: Substitute the coefficients into the quadratic formula

Substitute the coefficients \( a = 1 \), \( b = -1 \), and \( c = -1 \) into the quadratic formula: \[ u = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(1)(-1)}}{2(1)} \]

Step 3: Simplify the expression

Simplify the expression inside and under the square root: \[ u = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \].

Step 4: Calculate the roots

Finally, calculate the roots of the equation: \[ u = \frac{1 + \sqrt{5}}{2} \quad or \quad u = \frac{1 - \sqrt{5}}{2} \]

Key Concepts

Solving Quadratic Equations

Quadratic equations are a staple in algebra and come in the form of \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero. To solve such equations, one can factorize, complete the square, or, most commonly, use the quadratic formula. The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides a systematic solution by substituting the coefficients of the quadratic equation directly into the formula.

Let's say we have a quadratic equation \( u^2 - u - 1 = 0 \). Following step by step, we identify our coefficients \( a=1 \) (the number before \( u^2 \) which is: \( 1 \) in this case), \( b=-1 \) (the number before \( u \) which is: \( -1 \) here), and \( c=-1 \) (the constant term or the number without \( u \) which is: \( -1 \) here). By substituting these values into the quadratic formula, we can solve for \( u \) which yields the solution to the quadratic equation.

Quadratic Roots

The term 'roots' in the context of quadratic equations refers to the values of \( x \) (or any variable used) that satisfy the equation \( ax^2 + bx + c = 0 \). These values make the equation true when substituted into it. The quadratic formula provides these roots by incorporating the discriminant \( \sqrt{b^2 - 4ac} \), which determines the nature of the roots.

The discriminant can tell us if the roots are real and distinct, real and equal, or complex. A positive discriminant indicates two real distinct roots, zero discriminant indicates a single repeated real root, and a negative discriminant implies complex roots. For the quadratic equation \( u^2 - u - 1 = 0 \), the discriminant is \( 1^2 - 4(1)(-1) = 1 + 4 = 5 \), a positive number, hence it has two distinct real roots given by \( \frac{1 + \sqrt{5}}{2} \) and \( \frac{1 - \sqrt{5}}{2} \).

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two algebraic expressions and often contain one or more variables. They range from simple linear equations to more complex forms such as quadratic equations, polynomial equations, and beyond.

Understanding how to manipulate these equations is key to finding their solutions. Equations like \( u^2 - u - 1 = 0 \) are solved using various algebraic techniques, and the quadratic formula is a powerful tool for solving second-degree equations (quadratic). An essential aspect of algebra is recognizing patterns and learning the methods suitable for different kinds of equations, which alleviates the intimidation often associated with solving algebraic equations.