MathProblemSolver101

Well, let's see.

We can expand and simplify the product: $\left(\frac{a + b}{ab}\right)\left(\frac{a - b}{ab}\right) = \frac{a^2 - b^2}{a^2 b^2}$

We could try and make a negative with them, but $b$ is always positive, so that doesn't work.

Because of the squares, the numerator will always be nonnegative(the least value is $(-2)^2 - 2^2 = 0$) and the denominator always positive. So the least positive value is when the numerator is 0, which gives 0 as the solution.

Equation of a parabola is $y = a(x-h)^2 + k$

The vertex is $(h,k)$

We plug in and get $y = a(x-12)^2 - 4$

Plug in the coordinates: $0 = a(6^2) - 4$

$0 = 36a - 4$

$a = \frac{1}{9}$

The equation is now $y = \frac{1}{9}(x - 12)^2 - 4$

The other x-intercept is when $y=0,$ so we have $4 = \frac{1}{9}(x-12)^2$

$36 = x^2 - 24x + 144$

$x^2 - 24x + 108 = 0$

$r_1 + r_2 = 24$

$r_1 r_2 = 108$

We know that one root is $18,$ so we have $18 + r_2 = 24$ giving the answer to be $(24,0).$ The x-coordinate of that is $24.$

This happens when $x = 64, x^2 = 64, x^3 = 64, x^4 = 64$ because then the denominator will be $0.$

Thus the answer is $4.$

If you are looking for the values, they are $64, 8, 4, 2.$

Let $\gamma = \frac{1}{a}$ and $\theta = \frac{1}{b}.$

We have $\gamma x + \theta y = 1$

We have $\gamma \cdot 1 + \theta \cdot 11 = 1$

We also have $\gamma \cdot 4 + \theta \cdot 8 = 1$

Multiplying first equation by $4,$ we have $4 \gamma + 44 \theta = 4$

Subtracting, we have $36 \theta = 3$

$\theta = \frac{1}{12}$

$\gamma = \frac{11}{12}$

Thus, $a = \frac{1}{\gamma} = \frac{1}{\frac{11}{12}} = \boxed{\frac{12}{11}}$

$(4+1)(8+1)(2+1)(2+1) = 5 \times 9 \times 3 \times 3 = 45 \times 9 = 450 - 45 = \boxed{405}$

Let's see the equation of the line first. Choose two arbitrary points, say $(0,1), (1,0).$ We find the slope to be $1.$ The y-intercept is also $1,$ so our equation is $y = x + 1$(if it wasn't already trivial.)

We see that it cannot be less, and the line is straight not dotted, so the equation is $y \geq x + 1.$

$x^2 - 7x - 44 = 0$

$(x-11)(x+4) = 0$

$x = 11; x = -4.$