( a-4)²-2a(5a-4) =-⅓

Let's simplify the given expression step by step.

First, let's expand the expression (a - 4)²: (a - 4)² = (a - 4)(a - 4) = a² - 8a + 16

Next, let's simplify the expression -2a(5a - 4): -2a(5a - 4) = -10a² + 8a

Now, we can substitute these simplifications back into the original expression: (a - 4)² - 2a(5a - 4) = (a² - 8a + 16) - (-10a² + 8a)

Now, let's distribute the negative sign inside the parentheses: = a² - 8a + 16 + 10a² - 8a

Next, let's combine like terms: = (a² + 10a²) + (-8a - 8a) + 16 = 11a² - 16a + 16

To find the value of a for which this expression is equal to -⅓, we can set it equal to -⅓ and solve for a:

11a² - 16a + 16 = -⅓

Now we have a quadratic equation. Let's solve it.

Multiply both sides of the equation by 3 to remove the fraction: 33a² - 48a + 48 = -1

Move -1 to the left side of the equation: 33a² - 48a + 49 = 0

Unfortunately, in this case, the quadratic equation does not factor nicely. We can use the quadratic formula to find the solutions for a:

a = (-b ± √(b² - 4ac)) / 2a

In this case, a = 33, b = -48, and c = 49.

a = (-(-48) ± √((-48)² - 4(33)(49))) / (2 * 33) a = (48 ± √(2304 - 6468)) / 66 a = (48 ± √(-4164)) / 66

Since the discriminant (√(2304 - 6468)) is negative, there are no real solutions for a. Therefore, the equation cannot be equal to -⅓.