$\displaystyle \small \small \frac{1}{x}=\frac{x+1}{6}$

Cross multiply.

$\displaystyle \small 6=x(x+1)$

$\displaystyle \small 6=x^2+x$

Set the equation equal to zero.

$\displaystyle \small 0=x^2+x-6$

Factor to find the roots of the polynomial.

$\displaystyle 3*-2=-6$ and $\displaystyle 3+(-2)=1$

$\displaystyle \small 0=(x+3)(x-2)$

$\displaystyle \small 0=x+3; x=-3$

$\displaystyle \small 0=x-2; x=2$

In order to evaluate $\displaystyle (2x+3)^{2}$ one needs to multiply the expression by itself using the laws of FOIL.  In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

$\displaystyle \dpi{100} (2x+3)^{2}=(2x+3)(2x+3)$

Multiply terms by way of FOIL method.

$\displaystyle =(2x*2x)+(2x*3)+(3*2x)+(3*3)$

Now multiply and simplify.

$\displaystyle =4x^{2}+6x+6x+9$

$\displaystyle \rightarrow 4x^{2}+12x+9$

To solve our given equation, we need to use FOIL (First, Outer, Inner, Last).

$\displaystyle (x+7)(x-2)=x^2-2x+7x-14$

Combine like terms.

$\displaystyle (x+7)(x-2)=x^2+5x-14$

Remember FOIL stands for First Outer Inner Last.

$\displaystyle (x+1)(x-3)=x^2+1x-3x-3$

Combine like terms to get $\displaystyle x^2-2x-3$.

The formula for the discriminant is:

$\displaystyle b^2 - 4ac$

$\displaystyle =(-40)^2 - 4(4)(25)$

$\displaystyle =1200^{}$

Since the discriminant is positive and not a perfect square, there are $\displaystyle 2$ irrational roots.

The formula for the discriminant is:

$\displaystyle b^2 - 4ac$

$\displaystyle =(6)^2 - 4(2)(5)$

$\displaystyle =-4$

Since the discriminant is negative, there are $\displaystyle 2$ imaginary roots.

The formula for the discriminant is:

$\displaystyle b^2 - 4ac$

$\displaystyle =(-8)^2 - 4(4)(13)$

$\displaystyle =-144$ 

Since the discriminant is negative, there are $\displaystyle 2$ imaginary roots.

In general, the discriminant is $\displaystyle b^{2}-4ac$.

In this particual case $\displaystyle a=2, b=3\; and\; c=-5$.

Plug in these three values and simplify: $\displaystyle (3)^{2}-4(2)(-5)= 49$

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of $\displaystyle x$, and the product is the integer.

Sum: $\displaystyle (\frac{3}{4})+(\frac{1}{3})=\frac{13}{12}$

Product: $\displaystyle (\frac{3}{4})\cdot (\frac{1}{3})=\frac{1}{4}$

Subtract the sum and add the product.

The equation is:

$\displaystyle x^2 - \frac{13}{12}x+\frac{1}{4}=0$

Multiply the equation by $\displaystyle 12$:

$\displaystyle 12x^2+-13x+3=0$

To write an equation, find the sum and product of the roots. The sum is the negative coefficient of $\displaystyle x$, and the product is the integer.

Sum: $\displaystyle (2 + \sqrt{3})+(2 - \sqrt{3})=4$

Product: $\displaystyle (2 + \sqrt{3})\cdot (2 - \sqrt{3})=4-3=1$

Subtract the sum and add the product.

The equation is:

$\displaystyle x^2 -4x+1=0$