Problem 13 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(x^{2}+1=0\)

Step-by-Step Solution

Answer

Sum of the roots is 0, product of the roots is 1.

Step 1: Identify the coefficients

For a quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients. Here, we have \( x^2 + 1 = 0 \), which can be rewritten as \( 1x^2 + 0x + 1 = 0 \). Thus, \( a = 1 \), \( b = 0 \), and \( c = 1 \).

Step 2: Use the sum of the roots formula

The sum of the roots for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( -\frac{b}{a} \). For our equation, \( a = 1 \) and \( b = 0 \), so the sum of the roots is \( -\frac{0}{1} = 0 \).

Step 3: Use the product of the roots formula

The product of the roots for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \frac{c}{a} \). For our equation, \( a = 1 \) and \( c = 1 \), so the product of the roots is \( \frac{1}{1} = 1 \).

Key Concepts

Sum of Roots in Quadratic Equations

When dealing with a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots can be easily calculated using the formula: \( -\frac{b}{a} \). This formula comes from Viète's formulas, which relate the coefficients of the polynomial to sums and products of its roots.

• "Roots" are the values of \( x \) that satisfy the equation. In simpler terms, these are the values that make the equation equal to zero.
• The coefficient \( a \) is the number in front of \( x^2 \), and \( b \) is the coefficient in front of \( x \). "This formula tells us how these coefficients relate to the sum of the roots."

Using the equation \( x^2 + 1 = 0 \) as an example, we can identify that \( a = 1 \) and \( b = 0 \). Therefore, the sum of the roots is calculated as \( -\frac{0}{1} = 0 \).
This means that if we were to find the individual roots and add them together, the total would remain 0.

Product of Roots in Quadratic Equations

The product of the roots formula for a quadratic equation \( ax^2 + bx + c = 0 \) is \( \frac{c}{a} \). This is also derived from Viète's formulas. The formula gives a simple relationship between the constant term \( c \) and the leading coefficient \( a \) with the roots.

• The constant term \( c \) is the number without any \( x \) term attached to it.
• This product calculation tells us how the leading and constant coefficients influence the multiplication of the roots."

Applying this concept to our equation \( x^2 + 1 = 0 \), we identify \( a = 1 \) and \( c = 1 \). Thus, the product of the roots is \( \frac{1}{1} = 1 \).
In simple terms, if you multiply both roots together, you should get 1, demonstrating how the constants of the equation can determine this important characteristic even without solving the equation for its roots.

Complex Numbers in Quadratic Equations

When dealing with the equation \( x^2 + 1 = 0 \), an interesting aspect arises: the roots are actually complex numbers. Complex numbers are numbers that have both a real and an imaginary part and can be very useful in mathematics when real solutions do not exist.

• An imaginary number is usually noted with \( i \), which is defined as \( i = \sqrt{-1} \).
• When a quadratic has no real roots, it often means it will have complex roots instead."

For example, to solve \( x^2 + 1 = 0 \), subtract 1 from both sides to get \( x^2 = -1 \). When taking the square root of both sides, we find \( x = \pm i \). Thus, the roots are \( i \) and \( -i \).
By introducing complex numbers, we expand the types of problems we can solve and better understand uniquely challenging equations beyond those with real solutions.