Chapter 12

Simplify the expression. $$\sqrt{3} \cdot \sqrt{8}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{4 x+5}=x$$

State the basic axiom of algebra that is represented. $$x+(-x)=0$$

Decide whether the points are vertices of a right triangle. \((-2,0),(-1,0),(1,7)\)

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$b=11, c=61$$

Simplify the expression. $$(2+\sqrt{3})^{2}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x+6}-x=0$$

State the basic axiom of algebra that is represented. $$x\left(\frac{1}{x}\right)=1$$

Decide whether the points are vertices of a right triangle. \((2,0),(-2,2),(-3,-5)\)

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=12, b=35$$

Find the domain and the range of the function. $$y=3 \sqrt{x}$$

Solve \(x^{2}-3 x=8\) by completing the square. Solve the equation by using the quadratic formula. Which method did you find easier?

Simplify the expression. $$\sqrt{3}(5 \sqrt{3}-2 \sqrt{6})$$

Solve the equation. Check for extraneous solutions. $$x=\sqrt{x+12}$$

State the basic axiom of algebra that is represented. $$c d=d c$$

Find the midpoint between the two points. \((4,4),(-1,2)\)

Find the domain and the range of the function. $$y=\sqrt{x}$$

Solve by completing the square. $$x^{2}-2 x-18=0$$

Simplify the expression. $$\frac{4}{\sqrt{13}}$$

Solve the equation. Check for extraneous solutions. $$-5+\sqrt{x}=0$$

State the basic axiom of algebra that is represented. $$(x+y)+z=x+(y+z)$$

Find the midpoint between the two points. \((6,2),(2,-3)\)

Find the domain and the range of the function. $$y=\sqrt{x}-10$$

Solve by completing the square. $$x^{2}+14 x+13=0$$

Simplify the expression. $$\frac{3}{8-\sqrt{10}}$$

Solve the equation. Check for extraneous solutions. $$x=\sqrt{5 x+24}$$

Is subtraction closed for the positive real numbers? That is, if \(a\) and \(b\) are positive real numbers, must \((a-b)\) be a positive real number? Explain your thinking.

Find the midpoint between the two points. 12\. \((-5,3),(-3,-3)\)

Find the domain and the range of the function. $$y=\sqrt{x}+6$$

Solve by completing the square. $$3 x^{2}+4 x-1=0$$

Simplify the expression. $$\frac{6}{\sqrt{10}}$$

Find \(a\) if the geometric mean of 12 and \(a\) is \(6 .\)

\(a=b \quad\) Given \(a c=b c \quad\) Multiplication axiom of equality \(c=d\) \(\begin{array}{ll}b c=b d & \frac{1}{2} \\ a c=b d & \frac{?}{2}\end{array}\) \(\begin{array}{lr}b c=b d & ? \\ a c=b d & ?\end{array}\)

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=3, b=4$$

Find the domain and the range of the function. $$y=\sqrt{x+5}$$

Solve by completing the square. $$3 x^{2}-7 x+6=0$$

Copy and complete the proof of the statement: For all real numbers \(a\) and \(b,(a+b)-b=a\) SUPPLYING REASONS Copy and complete the proof of the statement: For all real numbers \(a\) and \(b,(a+b)-b=a\)

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(2,0),(8,-3)$$

Find the domain and the range of the function. $$y=\sqrt{x-10}$$

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$x^{2}-x-2=0$$

Show whether the expression is a solution of the equation. $$x^{2}-48=0 ;-4 \sqrt{3}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-9=0$$

Prove the theorem. (Use the basic axioms of algebra and the definition of subtraction given in Example \(1 .\) ) If \(a\) and \(b\) are real numbers, then \(a-b=-b+a\)

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(2,-8),(-3,3)$$

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$b=3, c=7$$

Sketch the graph of the function. $$y=3 \sqrt{x}$$

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$3 x^{2}+17 x+10=0$$

Show whether the expression is a solution of the equation. $$x^{2}-12 x+5=0 ; 6+\sqrt{31}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-1=0$$

Prove the theorem. (Use the basic axioms of algebra and the definition of subtraction given in Example \(1 .\) ) If \(a, b,\) and \(c\) are real numbers, then \((a-b)-c=a-(b+c)\)