Chapter 1: Fundamentals

Find the number of real solutions of the equation $3x^2=6x-9$ by using discriminant.

Simplify the fractional expression.

$\frac{{\left(1-x^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+x^2{\left(1-x^2\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{1-x^2}$

Anthony leaves Kingstown at $\text{2:00\hspace{0.33em}P.M.}$and drives to Queensville, $\text{160mi}$distant at $\text{45\hspace{0.33em}mi}/\text{h}$. At $\text{2:15\hspace{0.33em}P.M.}$Helen leaves Queensville and drives to Kingstown at $\text{40\hspace{0.33em}mi}/\text{h}$. At what time do they pass each other on the road?

Conjugate Recall that the symbol$\overline{z}$represents the complex conjugate of z. If$z=a+bi\text{and}w=c+di$, show that each statement is true.

$z-\overline{z}\text{isapureimaginarynumber.}$

Use a Special Factoring Formula to factor the expression

$1+1000y^3$

Solve the absolute value inequality. Express the answer using interval notation and graph the solution set.

$\left|5x-2\right|<8$

Using Slopes Verify the given geometric property.

Use slopes to determine whether the given points are collinear (lie on a line).

(a)$\left(1,1\right),\left(3,9\right),\left(6,21\right)$

(b) $\left(-1,13\right),\left(1,7\right),\left(4,15\right)$are vertices of a rectangle.

Perpendicular Bisector Find an equation of the perpendicular bisector of the line segment joining the points$A\left(1,4\right)$ and$B\left(7,-2\right)$

Solve the absolute value inequality. Express the answer using interval notation and graph the solution set.

$\left|3x-2\right|\ge 5$

Simplify the fractional expression.

$\frac{3{\left(1+x\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-x{\left(1+x\right)}^{-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{{\left(1+x\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$