- The Quadratic Formula TEKS FOCUS TEKS ()(F) Solve quadratic and square root equations. TEKS ()(G) Display, eplain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Additional TEKS ()(A), ()(F) VOCABULARY Discriminant The discriminant of a quadratic equation in the form a + b + c = is the value of the epression b - ac. Quadratic Formula The Quadratic Formula is -b { b - ac a. It gives the solutions to the quadratic equation a + b + c =. Justify eplain with logical reasoning. You can justify a mathematical argument. Argument a set of statements put forth to show the truth or falsehood of a mathematical claim 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. Key Concept The Quadratic Formula To solve the quadratic equation a + b + c =, use the Quadratic Formula. = - b { b - ac a 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 + a b + ( a) b = ( a) b - a c Rewrite so all terms containing are on one side. Complete the square. ( + a) b = b - ac a Factor the perfect square trinomial. Also, simplify. + b a = { b - ac a b =- a { b - ac a = - b { b - ac a Find square roots. Solve for. Also, simplify the radical. Lesson - The Quadratic Formula
The discriminant of a quadratic equation in the form a + b + c = is the value of the epression b - ac. = - b { b - ac d discriminant a Value of the Discriminant Key Concept Discriminant Key Concept 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 Quadratic Formula What are the solutions? Use the Quadratic Formula. Should you write the equation in standard form? Yes; write the equation in standard form to identify a, b, and c. 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. continued on net page PearsonTEXAS.com
Problem continued Why is there only one solution? Because if you add or subtract zero you get the same number. 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= Problem Applying the Quadratic Formula TEKS Process Standard ()(A) 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. 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. = - { - (-)(-) (-) = - { -. 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.......... Lesson - The Quadratic Formula
Problem TEKS Process Standard ()(G) 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. Problem Using the Discriminant to Solve a Problem STEM 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? What value should you substitute for h? You are trying to determine whether the ball will reach ft. Replace h with. h =-t + t + =-t + t + Substitute for h. =-t + t - Write the equation in standard form. a =-, b =, c =- Find the values of a, b, and c. b - ac = - (-)( - ) Evaluate the discriminant. = - =- The discriminant is negative. The equation =-t + t + has no real solutions. The golf ball will not reach a height of feet. PearsonTEXAS.com
ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Solve each equation using the Quadratic Formula. For additional support when completing your homework, go to PearsonTEXAS.com.. - + =. + + =. + =. ( - ) =. ( - ) =-. + =. Apply Mathematics ()(A) 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 $?. Apply Mathematics ()(A) 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.. + + =. - - =. - + - =. + - =. - + =. - + =. - + =. + =-. ( + ) =-. 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?. Create Representations to Communicate Mathematical Ideas ()(E) Summarize how to use the discriminant to analyze the types of solutions of a quadratic equation. Solve each equation using any method. When necessary, round real solutions to the nearest hundredth.. - - =. - - =. + - =. = -. =. + =. - + =. - - =. = -. Apply Mathematics ()(A) The 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. Lesson - The Quadratic Formula
. Apply Mathematics ()(A) 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 = - +. Analyze Mathematical Relationships ()(F) 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 Write a quadratic equation with the given solutions.. +, -. - +, - -. - +, - -. Use the Quadratic Formula to prove each statement. a. The sum of the solutions of the quadratic equation a + b + c = is - a b. b. The product of the solutions of the quadratic equation a + b + c = is c a. TEXAS Test Practice. 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 + B(, ) y Ú -. The graph of the system of inequalities μ is shown at the right. Ú A(, ) y Ú What is the maimum value of the function P = - y for the (, y) pairs in the bounded region shown? O C(, ) PearsonTEXAS.com