The Quadratic Formula

- The Quadratic Formula Content Standard Reviews A.REI..b Solve quadratic equations by... the quadratic formula... Objectives To solve quadratic equations using the Quadratic Formula To determine the number of solutions by using the discriminant To make a smile restrict your domain to K K. MATHEMATICAL PRACTICES On this happy face, what quadratic function graphs a smile that crosses the -ais twice? touches the -ais once? misses the -ais completely? Copy and show each completed face on graph paper. Eplain why each mouth meets the given condition. y AMIC I V I E S I T Dynamic Activity ERoots of a Quadratic S Lesson Vocabulary Quadratic Formula discriminant Another way to solve a quadratic equation a b c is by completing a square and then factoring. Essential Understanding You can solve a quadratic equation a b c in more than one way. In general, you can find a formula that gives values of in terms of a, b, and c. Here s how to solve a b c to get the Quadratic Formula. a b c a b a c Divide each side by a. b a c a b a Q b a R Q b a R c a Rewrite so all terms containing are on one side. Complete the square. Q b a R b ac a Factor the perfect square trinomial. Also, simplify. b a b ac Ä a b a "b ac a b "b ac a Find square roots. Solve for. Also, simplify the radical. Chapter Quadratic Functions and Equations

Key Concept The Quadratic Formula To solve the quadratic equation a b c, use the Quadratic Formula. b "b ac a Should you write the equation in standard form? Yes; write the equation in standard form to identify a, b, and c. Why is there only one solution? Because if you add or subtract zero you get the same number. Problem Using the Quadratic Formula What are the solutions? Use the Quadratic Formula. A Write in standard form. a, b, c Find the values of a, b, and c. b "b ac a () "() ()() ()!! or! Write the Quadratic Formula. Substitute for a, b, and c. Check Use a graphing calculator to graph y. The -intercepts are about. <! and. <!, as epected. B a, b, c Find the values of a, b, and c. " ()() ()!! Substitute into b "b ac a. Zero X=. Y= Got It?. What are the solutions? Use the Quadratic Formula. a. b. Lesson - The Quadratic Formula

Does it make sense that two different prices can yield the same profit? Yes. You can generate a given profit either by selling many CDs at a low price, or fewer CDs at a high price. Problem Applying the Quadratic Formula Fundraising Your school s jazz band is selling CDs as a fundraiser. The total profit p depends on the amount that your band charges for each CD. The equation p models the profit of the fundraiser. What is the least amount, in dollars, you can charge for a CD to make a profit of $? p Substitute for p. Write the equation in standard form. a, b, c Find the values of a, b, and c. " ()() ()! <. or <. Substitute into b "b ac. a Use a calculator. The least amount you can charge is $. for each CD to make a profit of $. The answer is... Got It?. a. In Problem, what is the least amount you can charge for each CD to make a $ profit? b. Reasoning Would a negative profit make sense in this problem? Eplain. A quadratic equation can have two real solutions ( ), one real solution ( ), or no real solutions ( ). In the Quadratic Formula, the value under the radical symbol, b ac, tells you how many real-number solutions eist. In Problem (a), b ac. and there are two real solutions. In Problem (b), b ac and there is only one real solution. Key Concept Discriminant The discriminant of a quadratic equation in the form a b c is the value of the epression b ac. b "b ac a d discriminant Chapter Quadratic Functions and Equations

Value of the Discriminant Discriminants and Solutions of Quadratic Equations Number of Solutions for a b c -intercepts of Graph of Related Function y a b c b ac. two real solutions two -intercepts b ac one real solution one -intercept b ac, no real solutions no -intercepts Problem Using the Discriminant What is the number of real solutions of? Find the values of a, b, and c. a, b, c Evaluate b ac. Interpret the discriminant. b ac () ()() The discriminant is positive. The equation has two real solutions. Got It?. What is the number of real solutions of each equation? a. b. Lesson - The Quadratic Formula

What value should you substitute for h? You are trying to determine whether the ball will reach ft. Replace h with. Problem Using the Discriminant to Solve a Problem Projectile Motion You hit a golf ball into the air from a height of in. above the ground with an initial vertical velocity of ft/s. The function h t t models the height, in feet, of the ball at time t, in seconds. Will the ball reach a height of ft? h t t t t t t a, b, c Substitute for h. Write the equation in standard form. Find the values of a, b, and c. b ac ()Q R Evaluate the discriminant. STEM The discriminant is negative. The equation t t has no real solutions. The golf ball will not reach a height of feet. Got It?. Reasoning Without solving an equation, will the golf ball in Problem reach a height of ft? Eplain. Lesson Check Do you know HOW? Solve each equation using the Quadratic Formula..... Find the discriminant of each quadratic equation. Determine the number of real solutions.... Do you UNDERSTAND? MATHEMATICAL PRACTICES. Reasoning For what values of k does the equation k have one real solution? two real solutions?. Error Analysis Your friend concluded that because two discriminants are equal, the solutions to the two equations are the same. Eplain your friend s error. Give an eample of two quadratic equations that disprove this conclusion.. Reasoning If one quadratic equation has a positive discriminant, and another quadratic equation has a discriminant equal to, can the two quadratic equations share a solution? Eplain why or why not. If so, give two quadratic equations that meet this criterion. Chapter Quadratic Functions and Equations

Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Solve each equation using the Quadratic Formula. See Problem.......... ( ). ( ). ( ).. Fundraising Your class is selling boes of flower seeds as a fundraiser. The total profit p depends on the amount that your class charges for each bo of seeds. The equation p. models the profit of the fundraiser. What s the smallest amount, in dollars, that you can charge and make a profit of at least $? See Problem.. Baking Your local bakery sells more bagels when it reduces prices, but then its profit changes. The function y. models the bakery s daily profit in dollars, from selling bagels, where is the price of a bagel in dollars. What s the highest price the bakery can charge, in dollars, and make a profit of at least $? Evaluate the discriminant for each equation. Determine the number of real solutions. See Problem............. ( ). Business Th e weekly revenue for a company is r p p, where p is the price of the company s product. Use the discriminant to find whether there is a price for which the weekly revenue would be $. See Problem. B Apply STEM. Physics Th e equation h t t models the height h in feet reached in t seconds by an object propelled straight up from the ground at a speed of ft>s. Use the discriminant to find whether the object will ever reach a height of ft.. Think About a Plan The area of a rectangle is in.. The perimeter of the rectangle is in. What are the dimensions of the rectangle to the nearest hundredth of an inch? How can you write an equation using one variable to find the dimensions of the rectangle? How can the discriminant of the equation help you solve the problem?. Writing Summarize how to use the discriminant to analyze the types of solutions of a quadratic equation. Lesson - The Quadratic Formula

Solve each equation using any method. When necessary, round real solutions to the nearest hundredth................ STEM. Air Pollution Th e function y... models the emissions of carbon monoide in the United States since, where y represents the amount of carbon monoide released in a year in millions of tons, and represents the year. a. How can you use a graph to estimate the year in which more than million tons of carbon monoide were released into the air? b. How can you use the Quadratic Formula to estimate the year in which more than million tons of carbon monoide were released into the air? c. Which method do you prefer? Eplain why.. Sports A diver dives from a m springboard. The equation f (t).t t models her height above the pool at time t in seconds. At what time does she enter the water? Without graphing, determine how many -intercepts each function has.. y. y.. y. y. y. y. y. y. y. Reasoning Determine the value(s) of k for which k has each type of solution. a. no real solutions b. eactly one real solution c. two real solutions. Use the discriminant to match each function with its graph. a. f () b. f () c. f () I. y II. y III. y O O O. a. Geometry Write an equation to find the dimensions of a square that has the same area as a circle with a radius of cm. b. Find the length of a side of the square, to the nearest hundredth centimeter. Chapter Quadratic Functions and Equations

C Challenge Write a quadratic equation with the given solutions..!,!.!!,.!!, Solve each equation.. u u. u u. u u. Use the Quadratic Formula to prove each statement. a. The sum of the solutions of the quadratic equation a b c is b a. b. The product of the solutions of the quadratic equation a b c is c a.. Eplain the meaning of the value "b ac a in terms of the graph of the standard quadratic function y a b c. Standardized Test Prep SAT/ACT. How many different real solutions are there for?. What is the y-value of the y-intercept of the quadratic function y ( )? y. What is the -value in the solution to the system e y? y # y $. The graph of the system of inequalities µ is shown at the right. $ y $ What is the maimum value of the function P y for the (, y) pairs in the bounded region shown? y A(, ) O B(, ) C(, ) Mied Review Solve each equation by completing the square. See Lesson -... y y. Simplify by combining like terms. See Lesson -.. z z z z. k k. y (y ) Get Ready! To prepare for Lesson -, do Eercises. Simplify each epression. See p... "(). ". " () Lesson - The Quadratic Formula