A quadratic equation has solutions of x=9 and x=−2. What is the quadratic equation?
It could be
(x-9)(x+2) = 0
or
x^2 - 7x - 18 = 0
Well, don't worry, I'm no square when it comes to quadratic equations! The equation that fits the bill with solutions of x=9 and x=-2 is (x - 9)(x + 2) = 0. And bingo, that's your quadratic equation! Keep on solving like a boss!
To find the quadratic equation when given the solutions, we can use the fact that the solutions can be used to factorize the equation.
So, if x = 9 and x = -2 are the solutions of the quadratic equation, then we can write the factors as (x - 9) and (x + 2).
To get the quadratic equation, we need to multiply these factors together:
(x - 9) * (x + 2)
Multiplying these factors gives us:
x^2 + 2x - 9x - 18
Simplifying, we get:
x^2 - 7x - 18
Therefore, the quadratic equation with solutions x = 9 and x = -2 is:
x^2 - 7x - 18 = 0
To find the quadratic equation, we can use the fact that the solutions of a quadratic equation can be found using the formula:
x = (-b ± √(b² - 4ac)) / 2a
Given that the solutions are x = 9 and x = -2, we can substitute these values into the formula:
For x = 9:
9 = (-b ± √(b² - 4ac)) / 2a
For x = -2:
-2 = (-b ± √(b² - 4ac)) / 2a
Now we have two equations. Since the quadratic equation has the same coefficients for both equations, we can substitute the values of x and rearrange the equation to solve for the coefficients.
Let's start by substituting x = 9 into the equation:
9 = (-b ± √(b² - 4ac)) / 2a
Multiply both sides of the equation by 2a:
18a = -2b ± √(b² - 4ac)
Now, let's substitute x = -2 into the equation:
-2 = (-b ± √(b² - 4ac)) / 2a
Multiply both sides of the equation by 2a:
-4a = -2b ± √(b² - 4ac)
Since the coefficients for both equations are the same, we can equate the two equations:
18a = -4a
-2b ± √(b² - 4ac) = √(b² - 4ac)
From the equation 18a = -4a, we can solve for a:
18a + 4a = 0
22a = 0
a = 0
Now, substitute a = 0 into one of the equations:
-2b ± √(b² - 4ac) = √(b² - 4ac)
Simplify the equation considering a = 0:
-2b = √(b²)
-2b = b
Solve for b:
-2b - b = 0
-3b = 0
b = 0
Now we have a = 0 and b = 0. Let's substitute these values back into one of the original equations, for example, x = 9:
9 = (-b ± √(b² - 4ac)) / 2a
With a = 0 and b = 0, we have:
9 = ± √(0 - 4(0)c) / 2(0)
9 = ± √(0) / 0
Since the square root of 0 is 0, we get:
9 = ± 0 / 0
At this point, the equation becomes indeterminate, meaning there is no valid quadratic equation that satisfies the given solutions of x = 9 and x = -2.