Quadratic equations are a fundamental part of algebra, often encountered in various fields of mathematics and science. These equations take the form of ax^2 + bx + c = 0, where ‘a,’ ‘b,’ and ‘c’ are constants, and ‘x’ represents the unknown variable. One powerful tool for solving quadratic equations is the quadratic formula. In this article, we will explore how to use the quadratic formula to solve the equation x^2 + 20 = 2x and determine the values of ‘x.’
Understanding the Quadratic Formula
The quadratic formula is a versatile tool used to find the solutions (also called roots) of any quadratic equation. It is given by:
x = (-b ± √(b² – 4ac)) / (2a)
In this formula:
- ‘a,’ ‘b,’ and ‘c’ represent the coefficients of the quadratic equation ax^2 + bx + c = 0.
- ‘√’ denotes the square root symbol.
- The symbol ‘±’ indicates that there are usually two solutions—one with a plus sign and one with a minus sign.
Solving x^2 + 20 = 2x
Now, let’s apply the quadratic formula to the equation x^2 + 20 = 2x. To do this, we need to identify the values of ‘a,’ ‘b,’ and ‘c’ from the equation:
- ‘a’ is the coefficient of x^2, which is 1 in this case.
- ‘b’ is the coefficient of x, which is -2 in this case.
- ‘c’ is the constant term, which is 20 in this case.
Plug these values into the quadratic formula:
x = (-(-2) ± √((-2)² – 4(1)(20))) / (2(1))
Now, simplify the formula:
x = (2 ± √(4 – 80)) / 2
x = (2 ± √(-76)) / 2
At this point, we notice that the expression under the square root (√(-76)) is a negative number. In the real number system, taking the square root of a negative number is not possible. Therefore, this quadratic equation has no real solutions.
Complex Solutions
However, quadratic equations can have solutions in the complex number system. Complex numbers have the form a + bi, where ‘a’ and ‘b’ are real numbers, and ‘i’ represents the imaginary unit (√(-1)). In this case, we can express the solutions as:
x = (2 ± √(-76)) / 2 x = (2 ± √76i) / 2 x = (2 ± 2√19i) / 2
Now, we can simplify further:
x = 1 ± √19i
So, the solutions to the equation x^2 + 20 = 2x in the complex number system are x = 1 + √19i and x = 1 – √19i.
Conclusion
In this article, we used the quadratic formula to solve the equation x^2 + 20 = 2x. While the solutions may not be real numbers, they exist in the complex number system as x = 1 + √19i and x = 1 – √19i. The quadratic formula is a powerful tool that enables us to find solutions to quadratic equations in a systematic way, whether the solutions are real or complex.
