Understanding Real Roots
A real root of a polynomial equation is a value of the variable that makes the polynomial equal to zero. In simpler terms, it is a value of x that satisfies the equation ax2 bx c 0 and makes it true. This value can be found using various methods, depending on the degree of the polynomial.
In the context of quadratic equations, the x-intercepts or the solutions to the equation are the real roots. These roots can be found using the quadratic formula or by factoring, if possible. The quadratic formula is given by:
x -b plusmn; radic;(b2 - 4ac) / (2a)
Finding Two Real Roots That Add Up to Zero
Example: Solving for Roots in a Quadratic Equation
Let's take an example with the equation x2 - 3x - 10 0.
Step 1: Identify the coefficients
a 1 b -3 c -10Step 2: Use the fact that the sum of the roots is -b/a
The sum of the roots is -(-3)/1 3. To make the sum zero, we set the roots to be -3/2 and 3/2 since -3/2 3/2 0.
Step 3: Solve for x in each case
Calculate x for both roots:
For x -3/2, substitute into the original equation:
x2 - 3x - 10 0
(-3/2)2 - 3(-3/2) - 10 0
9/4 9/2 - 10 0
9/4 18/4 - 40/4 0
-13/4 ≠ 0
For x 3/2, substitute into the original equation:
x2 - 3x - 10 0
(3/2)2 - 3(3/2) - 10 0
9/4 - 9/2 - 10 0
9/4 - 18/4 - 40/4 0
-59/4 ≠ 0
In both cases, the equation does not hold true, indicating that the roots are not real for this particular equation.
However, if the discriminant b2 - 4ac were positive, we would have two real roots that add up to zero.
Understanding Roots of Radicals
The n-th root of a number x is a number r such that rn x. This concept is fundamental in various areas of mathematics, including solving polynomial equations and understanding functions involving radicals.
For real numbers, the n-th root of a nonnegative number x has a unique nonnegative solution. This is because the equation rn x can have two real solutions (one positive and one negative) if n is even, but only one nonnegative solution if n is odd.
For instance, the square root of 16 is 4, as 42 16. The square root of 16 has another solution, -4, but in algebraic contexts, the principal (nonnegative) root is often used.
The notation radic;n x is used to represent the n-th root of x. For example, radic;2 x represents the square root of x.
Understanding the n-th root concept is crucial for solving equations and simplifying expressions involving radicals.