1.) To solve for x in the equation 2x(x - 12) = -40, we will first expand the equation:
2x(x - 12) = -40
2x^2 - 24x = -40
Next, we will rearrange the equation to bring all terms to one side, setting it equal to zero:
2x^2 - 24x + 40 = 0
This is now a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = -24, and c = 40. To solve for x, we can use the quadratic formula:
x = (-b Ā± ā(b^2 - 4ac)) / (2a)
Plugging in the values, we have:
x = (-(-24) Ā± ā((-24)^2 - 4(2)(40))) / (2(2))
x = (24 Ā± ā(576 - 320)) / 4
x = (24 Ā± ā256) / 4
x = (24 Ā± 16) / 4
This gives us two possible solutions:
x1 = (24 + 16) / 4 = 40 / 4 = 10
x2 = (24 - 16) / 4 = 8 / 4 = 2
Therefore, the solutions for x are x = 10 and x = 2.
2.) To solve for x in the equation 3x^2 + 2x = 12 - 7x, we will first rearrange the equation to bring all terms to one side and set it equal to zero:
3x^2 + 2x + 7x - 12 = 0
3x^2 + 9x - 12 = 0
Next, we can try factoring the quadratic equation. We look for two numbers that multiply to give -36 (the product of the coefficients of x^2 and the constant term) and add up to 9 (the coefficient of x). In this case, the numbers are 12 and -3:
3x^2 + 12x - 3x - 12 = 0
(3x + 12)(x - 1) = 0
Setting each factor equal to zero and solving for x, we have:
3x + 12 = 0 -> 3x = -12 -> x = -4
x - 1 = 0 -> x = 1
Therefore, the solutions for x are x = -4 and x = 1.
3.) To simplify the expression 8sqrt[-63], we can first factor -63 into its prime factorization. The prime factorization of 63 is 3 * 3 * 7, so we can write:
8sqrt[-63] = 8sqrt[(-1)(3)(3)(7)]
Next, we can rewrite the square root of a product as the product of the square roots:
8sqrt[(-1)(3)(3)(7)] = 8sqrt[-1] * sqrt[3] * sqrt[3] * sqrt[7]
Since the square root of -1 is denoted as "i" (the imaginary unit), we have:
8sqrt[-1] * sqrt[3] * sqrt[3] * sqrt[7] = 8i * ā3 * ā3 * ā7
Simplifying further:
8i * ā3 * ā3 * ā7 = 8i * 3 * ā7
Finally, multiplying the terms, we get:
8i * 3 * ā7 = 24iā7
Therefore, the simplified expression is 24iā7.
4.) To find the absolute value of the complex number 7 + 24i, we can use the formula for finding the modulus of a complex number:
|a + bi| = ā(a^2 + b^2)
Plugging in the values, we have:
|7 + 24i| = ā(7^2 + 24^2)
= ā(49 + 576)
= ā625
= 25
Therefore, the absolute value of the complex number 7 + 24i is 25.
5.) To simplify the expression [4 - 7i] + [2 + 3i], we will combine like terms by adding the real parts and the imaginary parts separately.
Real parts: 4 + 2 = 6
Imaginary parts: -7i + 3i = -4i
Therefore, the simplified expression is 6 - 4i.