User:MATH121/Sandbox

Find the equation of the line that intersects the line $$y=3x+2$$ at right angles when $$x=-2$$.

In problems 2.) and 3.) find all solutions $$x$$:

$$\cos\left(\frac{\pi}{4}\right)$$

$$\frac{2x^{2}-1}{2x^2+1}=\frac{1}{3}$$

Sketch four members of the following family of equations in $$x$$ and $$y$$ indexed with parameter $$k$$ for values of $$k=-1,0,1,2$$ and indicate on each graph the value of the parameter $$k$$ that corresponds to the graph. $$y=kx^{2}-5$$

For which values of $$m$$ does $$\left(x+3\right)\left(x^{2}+2mx-3m\right)=0$$ have 0, 1, 2, or 3 real-valued solutions?

Complete the square for the quadratic polynomial $$4x^{2}-24x+15$$.

Find the equation of the line that intersects the line $$y=3x+2$$ at right angles when $$x=2$$.

In problems 2.) and 3.) find all solutions $$x$$:

$$\sin\left(\frac{\pi}{4}x^{2}\right)=1$$

$$\frac{3x^2-1}{3x^2+1}=\frac{1}{3}$$

Sketch four members of the following family of equations in $$x$$ and $$y$$ indexed with parameter $$k$$ for values of $$k=-1,0,1,2$$ and indicate on each graph the value of the parameter $$k$$ that corresponds to the graph. $$y=-kx^{2}+5$$

For which values of $$m$$ does $$\left(x+2\right)\left(x^{2}+3mx-2m\right)=0$$ have 0, 1, 2, or 3 real-valued solutions?

Complete the square for the quadratic polynomial $$9x^{2}-54x+11$$.

Find the equations of the line that is the mirror image of the line $$y=4x-5$$ in the line $$x=-2$$.

In problems 2.) and 3.) find all solutions $$x$$:

$$\ln\left(8x+1\right)=2\ln\left(x+2\right)$$

$$\frac{1}{x+2}-\frac{1}{x+1}=\frac{1}{3x}$$

Sketch four members of the following family of equations in $$x$$ and $$y$$ indexed with parameter $$k$$ for values of $$k=-1,0,1,2$$ and indicate on each graph the value of the paramter $$k$$ that corresponds to the graph. $$2y=-kx+2$$

For which values of $$m$$ does $$\left(x-1\right)\left(x^{2}+2mx+2m\right)=0$$ have 0, 1, 2, or 3 real-valued solutions?

Use long division to write the rational function, $$\frac{3x^{3}+2x^{2}-4}{x^{2}-1}$$, in the form $$p\left(x\right)+\frac{q\left(x\right)}{r\left(x\right)}$$ where $$p\left(x\right)$$, $$q\left(x\right)$$, and $$r\left(x\right)$$ are polynomials and the degree of $$q\left(x\right)$$ is less than the degree of $$r\left(x\right)$$.

Find the equations of the line that is the mirror image of the line $$y=5x-3$$ in the line $$x=-3$$.

In proglems 2.) and 3.) find all solutions $$x$$:

$$\ln\left(3x+1\right)=2\ln\left(x+1\right)$$

$$\frac{1}{x+1}=\frac{1}{x+3}=\frac{1}{3x}$$

Sketch four members of the following family of equations in $$x$$ and $$y$$ indexed with parameter $$k$$ for values $$k=-1,0,1,2$$ and indicate on each graph the value of the parameter $$k$$ that corresponds to the graph. $$2y=kx-2$$

For which values of $$m$$ does $$\left(x-1\right)\left(x^{2}+4mx+4m\right)=0$$ have 0, 1, 2, or 3 real-valued solutions?

Use long division to write the rational function, $$\frac{4x^{3}+3x^{2}-4}{x^{2}+1}$$ in the form $$p\left(x\right)+\frac{q\left(x\right)}{r\left(x\right)}$$ where $$p\left(x\right)$$, $$q\left(x\right)$$, and $$r\left(x\right)$$ are polynomials and the degree of $$q\left(x\right)$$ is less than the degree of $$r\left(x\right)$$.