Chapter 1: Fundamentals

Find the real solutions of the quadratic equation $\frac{10}{x}-\frac{12}{x-3}+4=0$.

Rationalize the numerator.

$\frac{1-\sqrt{5}}{3}$

A 10ft long stem of bamboo is broken in such a way that its tip touches the ground 3ft from the base of the stem, as shown in the figure. What is the height of the break?

Find an equation of the circle that satisfies the given conditions.

Center at the origin; passes through$\left(4,7\right)$

Explain how to locate the point $\sqrt{2}$on a number line. Can you locate $\sqrt{5}$by a similar method? How can the circle is shown in the figure help us to locate $\pi$on a number line? List some other irrational numbers that you can locate on a number line.

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

${\mathbf{x}}^{5/2}\mathbf{-}{\mathbf{x}}^{1/2}$

Use the scientific notation, the Laws of exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data.

$\frac{1.295643\times {10}^9}{(3.610\times {10}^{-17})(2.511\times {10}^6)}$

Let $\max (a,b)$denote the maximum and$\max (a,b)$ denote the minimum of the real numbers a and b. For example$\max (2,5)=5$, and$\min (-1,-2)=-2$

a.Prove that$\max (a,b)=\frac{a+b+|a-b|}{2}$

b.Prove that$\min (a,b)=\frac{a+b-|a-b|}{2}$

Rationalize the numerator.

$\frac{\sqrt{3}+\sqrt{5}}{2}$

Find an equation of the circle that satisfies the given conditions.

Centre $(-1,5)$; passes through$(-4,-6)$