2(2
x
)
2
−7(2
x
)−4=0
To solve the equation 2(2x)² - 7(2x) - 4 = 0, let's start by simplifying:
2(2x)² is equal to 8x² (using the exponent rule).
-7(2x) is equal to -14x (using the distributive property).
Plugging these values back into the equation, we have:
8x² - 14x - 4 = 0.
Now, let's try to factorize the quadratic equation:
2(4x² - 7x - 2) = 0.
To factorize further, we need to determine two numbers that multiply to give -8 (product of the first and last coefficients) and add up to -7 (the coefficient of the middle term).
The two numbers are -8 and +1:
4x² - 8x + x - 2 = 0.
Factoring by grouping, we have:
(4x² - 8x) + (x - 2) = 0,
4x(x - 2) + (x - 2) = 0,
(4x + 1)(x - 2) = 0.
Now, we can set each factor equal to zero and solve for x:
4x + 1 = 0,
4x = -1,
x = -1/4.
x - 2 = 0,
x = 2.
So, the solutions to the equation 2(2x)² - 7(2x) - 4 = 0 are x = -1/4 and x = 2.
To solve the equation 2(2x)2 - 7(2x) - 4 = 0, we can use the quadratic formula.
The quadratic formula states that for an equation in the form of ax² + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b² - 4ac)) / (2a)
Comparing the given equation to the standard form, we have:
a = 2(2) = 4
b = -7(2) = -14
c = -4
Plugging these values into the quadratic formula, we get:
x = (-(-14) ± √((-14)² - 4(4)(-4))) / (2(4))
Simplifying further:
x = (14 ± √(196 + 64)) / 8
x = (14 ± √260) / 8
At this point, we can simplify the square root term:
x = (14 ± √4 * √65) / 8
x = (14 ± 2√65) / 8
Now, we can simplify the expression further by dividing both the numerator and denominator by 2:
x = (7 ± √65) / 4
So, the solutions to the equation 2(2x)2 - 7(2x) - 4 = 0 are:
x = (7 + √65) / 4
x = (7 - √65) / 4