Determining Values of k for No Real Solutions in a Quadratic Equation
Understanding when a quadratic equation does not have real solutions is crucial for various applications in mathematics and science. In this article, we will explore the values of k for which the quadratic equation x^2 kx 25 0 has no real solutions. This involves the use of the discriminant, a fundamental concept in algebra.
Introduction to the Quadratic Equation and the Discriminant
A quadratic equation is of the form ax^2 bx c 0, where a, b, and c are constants. The discriminant of this equation is defined as D b^2 - 4ac. The value of the discriminant determines the nature of the roots of the quadratic equation:
If D 0, the equation has two distinct real roots. If D 0, the equation has one real root (a repeated root). If D 0, the equation has no real roots (the roots are complex).Analysis of the Discriminant for Given Quadratic Equation
Consider the quadratic equation x^2 kx 25 0. Here, the coefficients are (a 1), (b k), and (c 25). Using the discriminant formula, we can determine the values of k for which the equation has no real solutions.
Substitute these values into the discriminant formula:
[D b^2 - 4ac k^2 - 4 cdot 1 cdot 25 k^2 - 100]For the equation to have no real solutions, the discriminant must be less than zero:
[k^2 - 100 0]Solving the Inequality
To solve the inequality (k^2 - 100 [(k 10)(k - 10) 0]
The roots of the equation (k^2 - 100 0) are (k -10) and (k 10). These roots divide the number line into three intervals: ((-infty, -10)), ((-10, 10)), and ((10, infty)). We need to test the sign of the expression ((k 10)(k - 10)) in these intervals:
Interval ((-infty, -10)): Choose (k -11), then ((-11 10)(-11 - 10) (-1)(-21) 21 0). Interval ((-10, 10)): Choose (k 0), then ((0 10)(0 - 10) (10)(-10) -100 0). Interval ((10, infty)): Choose (k 11), then ((11 10)(11 - 10) (21)(1) 21 0).From the tests, we see that the expression ((k 10)(k - 10) 0) holds in the interval ((-10, 10)). Therefore, the values of (k) for which the quadratic equation (x^2 kx 25 0) has no real solutions are:
[k in (-10, 10)]Conclusion and Summary
In summary, the quadratic equation (x^2 kx 25 0) has no real solutions when the discriminant is less than zero. By applying the discriminant formula and solving the resulting inequality, we found that k must lie in the interval ((-10, 10)).
To visualize this, consider the graph of the function y x^2 kx 25. The graph will not intersect the x-axis when k is within the range -10 k 10.
Images and graphs related to the solutions can be provided to make this concept clearer, but for now, the above analysis should suffice to understand the values of (k).