Lesson 2 Contents Example 1Two Real Solutions Example 2One Real Solution Example 3No Real Solution Example 4Estimate Roots Example 5Write and Solve an Equation

Example 2-1a Solveby graphing. Graph the related quadratic function The equation of the axis of symmetry is Make a table using x values around Then graph each point. x– f (x)f (x)0–4–6 –40

Example 2-1a From the table and the graph, we can see that the zeroes of the function are –1 and 4. Answer: The solutions of the equation are –1 and 4.

Example 2-1a CheckCheck the solutions by substituting each solution into the original equation to see if it is satisfied.

Example 2-1b Answer: –3 and 1 Solveby graphing.

Example 2-2a Solveby graphing. Write the equation in form. Add 4 to each side. Graph the related quadratic function x01234 f(x)f(x)41014

Example 2-2a Notice that the graph has only one x -intercept, 2. Answer: The equation’s only solution is 2.

Example 2-2b Solveby graphing. Answer: 3

Example 2-3a Number Theory Find two real numbers whose sum is 4 and whose product is 5 or show that no such numbers exist. ExploreLet one of the numbers. Then PlanSince the product of the two numbers is 5, you know that Original equation Distributive Property Add x 2 and subtract 4x from each side.

Example 2-3a x01234 f (x)52125 Solve You can solve by graphing the related function.

Example 2-3a Notice that the graph has no x -intercepts. This means that the original equation has no real solution. Answer:It is not possible for two numbers to have a sum of 4 and a product of 5. Examine Try finding the product of several numbers whose sum is 4.

Example 2-3b Number Theory Find two real numbers whose sum is 7 and whose product is 14 or show that no such numbers exist. Answer: no such numbers exist

Example 2-4a Solveby graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the axis of symmetry of the related function is x f(x)3–2–5–6–5–23

Example 2-4a The x -intercepts of the graph are between 0 and 1 and between 5 and 6. Answer:One solution is between 0 and 1 and the other is between 5 and 6.

Example 2-4b Solveby graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Answer:between 0 and 1 and between 3 and 4

Example 2-5a Royal Gorge Bridge The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h 0 is the initial height above the water in feet. We need to find t whenand Original equation

Example 2-5a Graph the related function using a graphing calculator. Adjust your window so that the x -intercepts are visible. Use the zero feature, [CALC], to find the positive zero of the function, since time cannot be negative. Use the arrow keys to locate a left bound for the zero and press. 2nd ENTER Then locate a right bound and presstwice. ENTER

Example 2-5a Answer: The positive zero of the function is approximately 8. It should take about 8 seconds for the marble to reach the surface of the water.

Example 2-5b Hoover Dam One of the largest dams in the United States is the Hoover Dam on the Colorado River, which was built during the Great Depression. The dam is feet tall. Suppose a marble is dropped over the railing from a height of 6 feet above the top of the dam. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h 0 is the initial height above the water in feet.

Example 2-5b Answer: about 7 seconds