Answer :
Sure, here are the roots of the given quadratic functions using the general formula or the factorization method, as appropriate:
Function 1:
f(x) = x² - 9
General Formula:
The general formula for the roots of a quadratic equation ax² + bx + c = 0 is:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 0, and c = -9. Substituting these values into the formula, we get:
x = (0 ± √(0² - 4 * 1 * -9)) / 2 * 1
x = (0 ± √36) / 2
x = (0 ± 6) / 2
Therefore, the roots of f(x) = x² - 9 are:
x1 = 3
x2 = -3
Factorization Method:
The quadratic equation can be factored as:
f(x) = (x + 3)(x - 3)
Therefore, the roots of f(x) = x² - 9 are:
x1 = -3
x2 = 3
Function 2:
f(x) = x² - 10x + 24
General Formula:
Using the general formula, we get:
x = (10 ± √(10² - 4 * 1 * 24)) / 2 * 1
x = (10 ± √(-64)) / 2
Since the square root of a negative number is an imaginary number, there are no real roots for this quadratic equation.
Factorization Method:
The quadratic equation can be factored as:
f(x) = (x - 4)(x - 6)
Therefore, the roots of f(x) = x² - 10x + 24 are:
x1 = 4
x2 = 6
Function 3:
f(x) = -3x² - 18x
General Formula:
Using the general formula, we get:
x = (18 ± √(18² - 4 * -3 * 0)) / 2 * -3
x = (18 ± √324) / -6
x = (18 ± 18) / -6
x = 0 or x = -3
Factorization Method:
The quadratic equation can be factored as:
f(x) = -3x(x + 6)
Therefore, the roots of f(x) = -3x² - 18x are:
x1 = 0
x2 = -6
Function 4:
f(x) = 8x² + 20
General Formula:
Using the general formula, we get:
x = (-20 ± √(20² - 4 * 8 * 20)) / 2 * 8
x = (-20 ± √(-640)) / 16
Since the square root of a negative number is an imaginary number, there are no real roots for this quadratic equation.
Factorization Method:
The quadratic equation cannot be factored using integer coefficients.
Function 5:
f(x) = 5x² - 15
General Formula:
Using the general formula, we get:
x = (15 ± √(15² - 4 * 5 * -15)) / 2 * 5
x = (15 ± √300) / 10
x = (15 ± 6√5) / 10
x = (3 ± 12√5) / 10
Therefore, the roots of f(x) = 5x² - 15 are:
x1 = (3 - 12√5) / 10
x2 = (3 + 12√5) / 10
Function 6:
f(x) = 10x² - 30x + 1
General Formula:
Using the general formula, we get:
x = (30 ± √(30² - 4 * 10 * 1)) / 2 * 10
x = (30 ± √840) / 20
x = (30 ± 6√14) / 20
x = (3 ± 3√14)
