By substituting \(\displaystyle u = x^{2}\) - and, subsequently, \(\displaystyle u^{2} =\left ( x^{2} \right )^{2} = x^{4}\) this can be rewritten as a quadratic equation, and solved as such:

\(\displaystyle x^{4} - 13x^{2} + 36 = 0\)

\(\displaystyle u^{2} - 13u + 36 = 0\)

We are looking to factor the quadratic expression as \(\displaystyle (u+?)(u+?)\), replacing the two question marks with integers with product 36 and sum \(\displaystyle -13\); these integers are \(\displaystyle -9,-4\).

\(\displaystyle (u-9)(u-4) = 0\)

Substitute back:

\(\displaystyle (x^{2}-9)(x^{2}-4) = 0\)

These factors can themselves be factored as the difference of squares:

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

Set each factor to zero and solve:

\(\displaystyle x-3 = 0 \Rightarrow x = 3\)

\(\displaystyle x-2= 0 \Rightarrow x = 2\)

\(\displaystyle x+2 = 0 \Rightarrow x = - 2\)

\(\displaystyle x+3 = 0 \Rightarrow x = - 3\)

The solution set is \(\displaystyle \left \{ -3, -2, 2, 3\right \}\).

\(\displaystyle 8x-5-4x=-x+10\) can be simplified to become

\(\displaystyle 4x-5=-x+10\)

Then, you can further simplify by adding 5 and \(\displaystyle x\) to both sides to get \(\displaystyle 5x=15\).

Then, you can divide both sides by 5 to get \(\displaystyle x=3\).

To solve for \(\displaystyle x\), you must first combine the \(\displaystyle x\)'s on the right side of the equation. This will give you \(\displaystyle \ 6x-1=9x+8\).

Then, subtract \(\displaystyle 8\) and \(\displaystyle 6x\) from both sides of the equation to get \(\displaystyle -9=3x\).

Finally, divide both sides by \(\displaystyle 3\) to get the solution \(\displaystyle x=-3\).

The first step is to distribute (multiply) the 2 through the parentheses:

\(\displaystyle 2(x+5) = -4\)

\(\displaystyle 2x + 10 = -4\)

Then isolate \(\displaystyle x\) on the left side of the equation. Subtract the 10 from the left and right side.

\(\displaystyle 2x + 10 = -4\)

\(\displaystyle 2x = -14\)

Finally, to isolate \(\displaystyle x\), divide the left side by 2 so that the 2 cancels out. Then divide by 2 on the right side as well.

\(\displaystyle 2x = -14\)

\(\displaystyle x = -7\)

You can verify this answer by plugging the \(\displaystyle -7\) into the original equation.

Combine like terms on the left side of the equation: \(\displaystyle 2x+2=3(x-1)\)

Use the distributive property to simplify the right side of the equation: \(\displaystyle 2x+2=3x-3\)

Next, move the \(\displaystyle x\)'s to one side and the integers to the other side: \(\displaystyle x=5\)

\(\displaystyle 12x-5(x-2)=45\)

Simplify the parenthesis:

\(\displaystyle 12x-5x+10=45\)

Combine the terms with x's:

\(\displaystyle 7x+10=45\)

Combine constants:

\(\displaystyle 7x=35\rightarrow \mathbf{x=5}\)

Solve the following equation when y is equal to four.

\(\displaystyle 14y-12x=200\)

To solve this equation, we need to plug in 4 for y and solve.

\(\displaystyle 14(4)-12x=200\)

\(\displaystyle 56-12x=200\)

\(\displaystyle -12x=144\)

\(\displaystyle x=-12\)

To solve, we must isolate x. In order to do that, we must first add 7 to both sides.

\(\displaystyle 3x-7+7=2+7\)

\(\displaystyle 3x=9\)

Next, we must divide both sides by 3.

\(\displaystyle \frac{3x}{3}=\frac{9}{3}\)

\(\displaystyle x=3\)

To find this line, first find the slope (m) between the two coordinate points. Then use the point-slope formula to find a line with that same slope passing through a particular point.

\(\displaystyle m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{2-10}{10-5}=-\frac{8}{5}\)

\(\displaystyle y-y_{1}=m(x-x_{1})\)

\(\displaystyle y-10=-\frac{8}{5}(x-5) \rightarrow y-10=-\frac{8}{5}x+8 \rightarrow \boldsymbol{y=-\frac{8}{5}x+18}\)

First distribute out each side of the equation.

\(\displaystyle 7(x-2)+10\) simplifies to \(\displaystyle 7x-4\).

Now for the right hand side, 

\(\displaystyle 8(x+1)\) becomes \(\displaystyle 8x+8\).

Now we equate both sides.

\(\displaystyle 7x-4 =8x+8\),

\(\displaystyle x=-12\)