What is the sum of all distinct real solutions to the equation
For quadratic equations set equal to a constant, first move all terms to one side so the equation equals 0, then look to factor quickly if the numbers are simple. Once factored, set each factor equal to 0 to find all real solutions, and pay close attention to what the question asks for—such as the sum or product of the solutions—rather than stopping after finding just one root.
Hints
Get everything on one side
Try moving the 6 to the left side so that one side of the equation becomes 0. This will give you a standard quadratic equation.
Factor the quadratic
Once you have , look for two numbers that multiply to and add up to so you can factor the expression.
Use the solutions correctly
After you find the two values of that satisfy the equation, remember the question is asking for the sum of these distinct real solutions, not just one of them.
Desmos Guide
Enter the equation in Desmos
Type y = x^2 - 5x on one line and y = 6 on another line. This graphs both sides of the original equation.
Find the intersection points
Look for the points where the parabola intersects the horizontal line . Note the -coordinates of these intersection points; these are the real solutions.
Compute the sum of the solutions
Add the -values of the intersection points (you can use Desmos’s calculator line for this) to get the sum of all distinct real solutions.
Step-by-step Explanation
Rewrite the equation in standard quadratic form
Start with the given equation:
Move 6 to the left side to set the equation equal to 0:
Now the equation is in standard quadratic form .
Factor the quadratic and find the solutions
Factor the quadratic expression .
We look for two numbers that multiply to and add to . Those numbers are and :
Set each factor equal to 0:
So the real solutions are and .
Add the distinct real solutions
The distinct real solutions are and .
Add them:
So, the sum of all distinct real solutions is .