\(\displaystyle -8x^2=-144\) Divide \(\displaystyle -8\) on both sides.

\(\displaystyle x^2=18\) Take the square root on both sides. Remember to account for a negative square root. Two negatives multiplied is a positive number.

\(\displaystyle x=\pm \sqrt{18}=\pm\sqrt{9}*\sqrt{2}=\pm3\sqrt{2}\)

\(\displaystyle \frac{x-8}{8}=\frac{4}{x+6}\) Cross-multiply.

\(\displaystyle (x-8)(x+6)=4*8\) Foil out the terms and simplify.

\(\displaystyle x^2-2x-48=32\) Subtract \(\displaystyle 32\) on both sides. 

\(\displaystyle x^2-2x-80=0\) We have a quadratic equation. We need to find two terms that multiply to \(\displaystyle -80\) and aso add to \(\displaystyle -2\).

\(\displaystyle (x-10)(x+8)\) Set them individualy equal to zero.

\(\displaystyle x-10=0\) Add \(\displaystyle 10\) to both sides. \(\displaystyle x=10\)

\(\displaystyle x+8=0\) Subtract \(\displaystyle 8\) on both sides. \(\displaystyle x=-8\)

We should still check the answers.

\(\displaystyle \frac{10-8}{8}=\frac{4}{10+6}\)  With simplifications, \(\displaystyle \frac{2}{8}=\frac{4}{16}=\frac{1}{4}\) . \(\displaystyle 10\) is good.

\(\displaystyle \frac{-8-8}{8}=\frac{4}{-8+6}\)  With simplifications, \(\displaystyle \frac{-16}{8}=\frac{4}{-2}=-2\) . \(\displaystyle -8\) is good.

Answers are \(\displaystyle 10, -8\).

\(\displaystyle 4x-15=145\) Add \(\displaystyle 15\) on both sides.

\(\displaystyle 4x=160\) Divide \(\displaystyle 4\) on both sides.

\(\displaystyle x=40\)

\(\displaystyle 3x-83=-11\) Add \(\displaystyle 83\) on both sides. 

\(\displaystyle 3x=72\) Divide \(\displaystyle 4\) on both sides.

\(\displaystyle x=24\)

\(\displaystyle 4\left | x\right |+93=145\) Subtract \(\displaystyle 93\) on both sides.

\(\displaystyle 4\left | x\right |=52\) Divide \(\displaystyle 4\) on both sides.

\(\displaystyle \left | x\right |=13\) Since it's absolute value, we need to accept both positive and negative answers.

\(\displaystyle x=\pm13\)

\(\displaystyle \frac{\sqrt{x}-6}{3}=8\) Multiply \(\displaystyle 3\) on both sides.

\(\displaystyle \sqrt{x}-6=24\) Add \(\displaystyle 6\) on both sides.

\(\displaystyle \sqrt{x}=30\) Square both sides to get rid of the radical.

\(\displaystyle x=30^2=900\)

\(\displaystyle 18(x+6)=4x\) Distribute the \(\displaystyle 18\) to each term in the parentheses.

\(\displaystyle 18x+108=4x\) Subtract \(\displaystyle 18x\) on both sides.

\(\displaystyle 108=-14x\) Divide \(\displaystyle -14\) on both sides.

\(\displaystyle x=\frac{-108}{14}=-\frac{54}{7}\)

\(\displaystyle (x+5)^2=289\) Take the square root on both sides. When you do that, you also need to consider both positive and negative values. Remember, two negatives multiplied create a positive number.

\(\displaystyle x+5=17\) Subtract \(\displaystyle 5\) on both sides.

\(\displaystyle x=12\)

\(\displaystyle -(x+5)=17\) Divide \(\displaystyle -1\) on both sides.

\(\displaystyle x+5=-17\) Subtract \(\displaystyle 5\) on both sides.

\(\displaystyle x=-22\) 

Answers are \(\displaystyle -22, 12\).

The first step will be to plug our given variable into the equation to get 

\(\displaystyle y=5\left ( 5\right )-25\).  

Then you do the multiplication first so it is now, 

\(\displaystyle \\(5*5=25) \\y=25-25\).  

Finally, subtract \(\displaystyle 25\) from \(\displaystyle 25\) to get \(\displaystyle y=0\).

A cubic polynomial has three zeroes, if a zero of degree \(\displaystyle n\) is counted \(\displaystyle n\) times. Since its lead term is \(\displaystyle x^{3}\), we know that, in factored form,

\(\displaystyle p(x) = (x-b_{1}) (x-b_{2}) (x-b_{3})\), 

where \(\displaystyle b_{1}\), \(\displaystyle b_{2}\), and \(\displaystyle b_{3}\) are its zeroes.

A polynomial with rational coefficients has its imaginary zeroes in conjugate pairs. Since \(\displaystyle p(x)\) is such a polynomial, then, since \(\displaystyle 2 + 3i\) is one of its zeroes, so is its complex conjugate, \(\displaystyle 2 -3i\). It has one other known zero, 2.

Therefore, we can set \(\displaystyle b_{1} = 2 + 3i\), \(\displaystyle b_{2} = 2 -3i\), \(\displaystyle b_{3} = 2\) in the factored form of \(\displaystyle p(x)\), and

\(\displaystyle p(x) = [x-(2+3i)] [x-(2-3i)](x-2)\)

To rewrite this, first multiply the first two factors with the help of the difference of squares pattern and the square of a binomial pattern:

\(\displaystyle [x-(2+3i)] [x-(2-3i)]\)

\(\displaystyle = [(x- 2) - 3i] [(x- 2 )+ 3i]\)

\(\displaystyle = (x- 2)^{2} - (3i)^{2}\)

\(\displaystyle = x^{2} - 4x+4 +9\)

\(\displaystyle = x^{2} - 4x+13\)

Thus,

\(\displaystyle p(x) = (x^{2} - 4x+13)(x-2)\)

Distributing:

\(\displaystyle = x^{2} (x-2)- 4x(x-2)+13 (x-2)\)

\(\displaystyle = x^{3}-2x^{2} - 4x^{2}+8x+13x - 26\)

\(\displaystyle = x^{3}-6x^{2} +21x - 26\)