Write a quadratic equation having the given solutions 14 and -9
By factoring, a quadratic equation with known zeroes can be expressed as:
(x-x1)(x-x2)=0
where x1 and x2 are the solutions to the quadratic equation.
Substitute the zeroes in the above equation and expand to give the quadratic equation.
To write a quadratic equation with the given solutions, we can use the fact that for a quadratic equation in standard form: ax^2 + bx + c = 0, the solutions are found by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Given the solutions x = 14 and x = -9, we can substitute these values into the equation to get two separate equations:
1) For x = 14:
14 = (-b ± √(b^2 - 4ac)) / 2a
2) For x = -9:
-9 = (-b ± √(b^2 - 4ac)) / 2a
To simplify the equations, we can eliminate the square root by squaring both sides of the equations:
1) (14)^2 = ((-b ± √(b^2 - 4ac)) / 2a)^2
196 = (b^2 - 4ac) / (4a^2)
2) (-9)^2 = ((-b ± √(b^2 - 4ac)) / 2a)^2
81 = (b^2 - 4ac) / (4a^2)
Now, we have a system of two equations with two variables (a, b, and c). We can solve these equations simultaneously to find the values of a, b, and c.
Let's solve the system of equations:
196 = (b^2 - 4ac) / (4a^2)
81 = (b^2 - 4ac) / (4a^2)
Simplifying both equations gives:
784a^2 = b^2 - 4ac
324a^2 = b^2 - 4ac
Since both expressions are equal to b^2 - 4ac, we can equate them:
784a^2 = 324a^2
Subtracting 324a^2 from both sides gives:
460a^2 = 0
Dividing both sides by 460 gives:
a^2 = 0
Taking the square root of both sides gives:
a = 0
Now that we have the value of a, we can substitute it back into one of the original equations to find b and c. Let's use equation 1) for convenience:
196 = (b^2 - 4(0)(c)) / (4(0)^2)
196 = b^2
Taking the square root of both sides gives:
b = ± 14
Therefore, the quadratic equation with the given solutions 14 and -9 is:
x^2 - 196 = 0