(x-5)(x-3)>K
Find all possible values of k
To find the possible values of k that satisfy the inequality, we need to solve the quadratic inequality (x-5)(x-3) > k.
Here's how you can approach it:
1. Expand the quadratic expression:
(x-5)(x-3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15
2. Rewrite the inequality:
x^2 - 8x + 15 > k
3. Move all terms to one side of the inequality:
x^2 - 8x + 15 - k > 0
4. To solve this inequality, you can use various methods, such as factoring, completing the square, or using the quadratic formula. However, in this case, it is easier to use a graphical approach.
5. Sketch the graph of the quadratic function y = x^2 - 8x + 15 - k.
The graph of a quadratic function is a parabola. By analyzing the shape of the parabola and its position relative to the x-axis, you can determine the possible values of k.
- If the parabola is completely above the x-axis, it means that y > 0 for all values of x. Therefore, there are no restrictions on k, and all real numbers are possible values.
- If the parabola intersects or touches the x-axis at any point(s), it means that there are certain values of k that make the inequality hold true. In this case, we need to find those values.
6. Find the x-intercepts of the parabola by setting y = 0:
x^2 - 8x + 15 - k = 0
Solve the quadratic equation to find the roots. These values of x will give us the critical points where the parabola intersects the x-axis.
7. Use the discriminant to determine the number of roots and their nature:
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
- If D > 0, the equation has two distinct roots and the parabola intersects the x-axis at two points.
- If D = 0, the equation has one real root and the parabola touches the x-axis at one point.
- If D < 0, the equation has no real roots and the parabola does not intersect the x-axis.
8. Analyze the results of step 7:
- If the discriminant is negative (D < 0), there are no real roots, which means the parabola does not intersect or touch the x-axis for any value of k. Therefore, all real numbers are possible values of k.
- If the discriminant is zero (D = 0), there is one real root. In this case, set the equation equal to zero and solve for x to find the critical point(s). Then, substitute the x-values into the original inequality to find the corresponding values of k that satisfy the inequality.
- If the discriminant is positive (D > 0), there are two real roots. Follow the same procedure as in the previous step to find the critical points and substitute them into the inequality to find the corresponding values of k.
By following these steps, you can find all possible values of k that satisfy the quadratic inequality (x-5)(x-3) > k.