Find the x-intercepts of f(x)=4x^2-20.
Give an exact answer using radicals as needed. Rationalize all denominators. Express complex numbers in terms of i.
The x intercept is where f(x) = 0
That would be where 4x^2 = 20, or in other words, where
x^2 = 5
Take the square root of both sides, remembering to use +/-. There should be no denominator to rationalize, so forget about that
To find the x-intercepts of the function f(x), you need to set f(x) equal to zero and solve for x.
Given the function f(x) = 4x^2 - 20, set f(x) equal to zero:
0 = 4x^2 - 20
To solve this quadratic equation, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = 0, and c = -20. Substituting these values into the quadratic formula:
x = (0 ± √(0^2 - 4(4)(-20))) / (2(4))
= (0 ± √(0 + 320)) / 8
= (0 ± √320) / 8
Simplifying further, we have:
x = ±√320 / 8
To express this answer with radicals, we can simplify the square root of 320:
√320 = √(16 * 20) = √16 * √20 = 4√20
Substituting this back into the equation for x:
x = ±(4√20) / 8
We can simplify by dividing both the numerator and denominator by 4:
x = ±√20 / 2
Rationalizing the denominator, we multiply both the numerator and denominator by √20:
x = ±(√20 * √20) / (2 * √20)
= ±√(20 * 20) / (2 * √20)
= ±√400 / 2
= ±20 / 2
= ±10
Therefore, the x-intercepts of the function f(x) = 4x^2 - 20 are x = -10 and x = 10.