Answer :
The equation 48x³ + 40x² - x - 3 = 0 has a real zero of x = -(3/4), and no other real solutions.
Here, we have,
To solve the equation 48x³ + 40x²- x - 3 = 0 and given that -(3/4) is a zero of the function, we can use synthetic division to divide the polynomial by (x + 3/4).
Using synthetic division:
-3/4 | 48 40 -1 -3
| 0 -36 18 4
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48 4 17 1
The result is a quotient of 48x² + 4x + 17 and a remainder of 1.
The equation can now be rewritten as:
48x² + 4x + 17 + 1 = 0
Simplifying:
48x² + 4x + 18 = 0
Since this quadratic equation does not factor easily, we can use the quadratic formula to find the remaining solutions:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 48, b = 4, and c = 18.
Plugging these values into the quadratic formula:
x = (-4 ± √(4² - 4(48)(18))) / (2(48))
x = (-4 ± √(16 - 3456)) / (96)
x = (-4 ± √(-3440)) / (96)
Since the discriminant (b² - 4ac) is negative, the equation has no real solutions.
The given zero of -(3/4) is the only real root of the equation.
Therefore, the equation 48x³ + 40x² - x - 3 = 0 has a real zero of
x = -(3/4), and no other real solutions.
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