5-Minute Check on Chapter 2 Transparency 3-1 5-Minute Check on Chapter 2 Evaluate 42 - |x - 7| if x = -3 Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green? Which of the following is a true statement Standardized Test Practice: A 8/4 < 4/8 B -4/8 < -8/4 C -4/8 > -8/4 D -4/8 > 4/8 Click the mouse button or press the Space Bar to display the answers.

Solving Quadratic Equations by Using the Quadratic Formula Lesson 10-4 Solving Quadratic Equations by Using the Quadratic Formula

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Objectives Solve quadratic equations by using the Quadratic formula Use the discriminant to determine the number of solutions for a quadratic equation

Vocabulary Quadratic formula – Discriminant –

Working Backwards Start with the answer “Undo” the operation that got you to the answer Keep “undoing” until you get back to the beginning

Example 1 Use two methods to solve Method 1 Factoring Original equation Factor Zero Product Property or Solve for x.

Example 1 cont Method 2 Quadratic Formula For this equation, Multiply.

Example 1 cont Add. Simplify. or Answer: The solution set is {–5, 7}.

Example 2 Solve by using the Quadratic Formula. Round to the nearest tenth if necessary. Step 1 Rewrite the equation in standard form. Original equation Subtract 4 from each side Simplify.

Example 2 cont Step 2 Apply the Quadratic Formula. Quadratic Formula a = 15, b = -8 and c = -4 Multiply, then Add. or

Example 2 cont Check the solutions by using the CALC menu on a graphing calculator to determine the zeros of the related quadratic function. Answer: The approximate solution set is {–0.3, 0.8}.

Example 3 Space Travel Two possible future destinations of astronauts are the planet Mars and a moon of the planet Jupiter, Europa. The gravitational acceleration on Mars is about 3.7 meters per second squared. On Europa, it is only 1.3 meters per second squared. Using the information and equation from Example 3 on page 548 in your textbook, find how much longer baseballs thrown on Mars and on Europa will stay above the ground than a similarly thrown baseball on Earth. In order to find when the ball hits the ground, you must find when H = 0. Write two equations to represent the situation on Mars and on Europa.

Example 3 cont Baseball Thrown on Mars Baseball Thrown on Europa These equations cannot be factored, and completing the square would involve a lot of computation.

Example 3 cont To find accurate solutions, use the Quadratic Formula. Since a negative number is not reasonable, use the positive solutions. Answer: A ball thrown on Mars will stay aloft 5.6 – 2.2 or about 3.4 seconds longer than the ball thrown on Earth. The ball thrown on Europa will stay aloft 15.6 – 2.2 or about 13.4 seconds longer than the ball thrown on Earth.

Example 4a State the value of the discriminant for . Then determine the number of real roots of the equation. and Simplify. Answer: The discriminant is –220. Since the discriminant is negative, the equation has no real roots.

Example 4b State the value of the discriminant for . Then determine the number of real roots of the equation. Step 1 Rewrite the equation in standard form. Original equation Add 144 to each side Simplify. Step 2 Find the discriminant. a = 1, b = 24 and c = 144 Simplify. Answer: The discriminant is 0. Since the discriminant is 0, the equation has one real root.

Example 4c State the value of the discriminant for . Then determine the number of real roots of the equation. Step 1 Rewrite the equation in standard form. Original equation Subtract 12 from each side Simplify. Step 2 Find the discriminant. a = 3, b =10 and c = -12 Simplify Answer: The discriminant is 244. Since the discriminant is positive, the equation has two real roots.

Summary & Homework Summary: Homework: The solutions of a quadratic equation in the form ax2 + bx + c = 0, where a ≠ 0, are given by the Quadratic Formula: Homework: pg -b ± √b² - 4ac x = ----------------------- 2a