a. 12x^2 = 17x - 6:
To solve this equation, we need to rearrange it into the standard form, which is a quadratic equation in the form of ax^2 + bx + c = 0.
12x^2 - 17x + 6 = 0
Next, we can factorize the equation, if possible. If not, we can use the quadratic formula to find the solutions.
To factorize, we need to find two numbers that multiply to give us 12 * 6 = 72 and add to give us -17.
The numbers are -8 and -9.
Therefore, the equation can be factored as follows:
(3x - 2)(4x - 3) = 0
Setting each factor equal to zero, we get:
3x - 2 = 0 -> 3x = 2 -> x = 2/3
4x - 3 = 0 -> 4x = 3 -> x = 3/4
So the solutions are x = 2/3 and x = 3/4.
b. 5x^2 = 6x – 4:
Again, let's rearrange the equation into the standard form:
5x^2 - 6x + 4 = 0
This equation cannot be easily factored, so we'll use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 5, b = -6, and c = 4. Substituting these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4 * 5 * 4)) / (2 * 5)
= (6 ± √(36 - 80)) / 10
= (6 ± √(-44)) / 10
Since we have a square root of a negative number, the solutions will be complex numbers. Thus, the equation has no real solutions.
c. x^6 = 6x^3 + 16 (real solutions):
To solve this equation, let's move all terms to one side and set the equation equal to zero:
x^6 - 6x^3 - 16 = 0
Since this is a sixth-degree polynomial equation, it cannot be easily factored or solved algebraically. We'll need to use numerical methods or graphical approaches to find the real solutions.
One possible numerical method is to use the bisection method or approximation algorithms to approximate the real solutions.
d. |2x - 3| - 7 = 4:
To solve this equation, let's consider the two cases based on the absolute value:
1. 2x - 3 = 4 + 7 => 2x - 3 = 11 => 2x = 14 => x = 7
2. -(2x - 3) - 7 = 4 => -2x + 3 - 7 = 4 => -2x - 4 = 4 => -2x = 8 => x = -4
Therefore, the solutions are x = 7 and x = -4.
e. 18x^2 - 2x - 8:
To solve this quadratic equation, let's apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 18, b = -2, and c = -8. Substituting these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4 * 18 * -8)) / (2 * 18)
= (2 ± √(4 + 576)) / 36
= (2 ± √580) / 36
Since the value under the square root (√580) is not a perfect square, the solutions will be irrational and can be expressed as (2 ± √580) / 36.
f. (3/5) = (x - 4) / (x + 2):
To solve this equation, let's cross-multiply and simplify:
3(x + 2) = 5(x - 4)
3x + 6 = 5x - 20
6 + 20 = 5x - 3x
26 = 2x
x = 13
Therefore, the solution is x = 13.
g. 15 - x - 1 = x:
To solve this equation, let's combine like terms:
14 - x = 2x
14 = 3x
x = 14/3
Therefore, the solution is x = 14/3.
h. e^(2x + 3) = 12:
To solve this equation, let's take the natural logarithm (ln) of both sides:
ln(e^(2x + 3)) = ln(12)
2x + 3 = ln(12)
Next, let's isolate x by subtracting 3 from both sides:
2x = ln(12) - 3
x = (ln(12) - 3) / 2
Therefore, the solution is x = (ln(12) - 3) / 2.
i. 4ln(2x + 1) = 10:
To solve this equation, let's divide both sides by 4:
ln(2x + 1) = 10/4
ln(2x + 1) = 5/2
Next, let's exponentiate both sides with the base e:
2x + 1 = e^(5/2)
Now, let's isolate x by subtracting 1 from both sides:
2x = e^(5/2) - 1
x = (e^(5/2) - 1) / 2
Therefore, the solution is x = (e^(5/2) - 1) / 2.
j. log(3x + 4) + log(x - 1) = log(x + 2):
To solve this equation, let's combine the logarithms using the properties of logarithms:
log((3x + 4)(x - 1)) = log(x + 2)
Since the logarithms are equal, we can drop the logarithm symbols:
(3x + 4)(x - 1) = x + 2
Now, let's expand and simplify:
3x^2 + x - 4x - 4 = x + 2
3x^2 - 4x - 4x - x - 4 - 2 = 0
3x^2 - 9x - 6 = 0
This equation cannot be easily factored, so we'll use the quadratic formula:
x = (-(-9) ± √((-9)^2 - 4 * 3 * -6)) / (2 * 3)
= (9 ± √(81 + 72)) / 6
= (9 ± √153) / 6
Therefore, the solutions are x = (9 + √153) / 6 and x = (9 - √153) / 6.
k. 2x^3 = 48:
To solve this equation, let's divide both sides by 2:
x^3 = 24
Next, let's take the cube root of both sides:
x = ∛24
Therefore, the solution is x = ∛24.
8. Inequality: 17x – 312 < 4:
To solve this inequality, let's isolate x:
17x < 4 + 312
17x < 316
Now, divide both sides of the inequality by 17:
x < 316 / 17
Therefore, the solution is x < 18.588.
9. Inequality: 7x - 3| < 4:
To solve this inequality, let's consider the two cases based on the absolute value:
1. -3(7x - 3) < 4 => -21x + 9 < 4 => -21x < -5 => x > 5/21
2. 7x - 3 < 4 => 7x < 7 => x < 1
Therefore, the solution to the inequality is x > 5/21 or x < 1.
Note: The exact solutions to the equations and inequalities can vary based on the provided details of the question. Please ensure that all necessary information is provided for accurate solutions.