5-8 T HE QUADRATIC F ORMULA Chapter 5 Quadratic Functions and Equations ©Tentinger

E SSENTIAL U NDERSTANDING AND O BJECTIVES Essential Understanding: you can solve quadratic equations in more than one way. In general you can find a formula that gives the values of x in terms of a, b, and c Objectives : Students will be able to: Solve quadratic equations using the quadratic formula Determine the number of solutions using the discriminant

I OWA C ORE CURRICULUM Algebra Reviews A.REI.4b. Solve quadratic equations in one variable. Solve quadratic equations by inspection taking square roots, completing the square, the quadratic formula and factoring, as appropriate tot eh initial form of the equation. Recognize then the quadratic formula gives complex solutions and write them as a±bi for real numbers a and b.

D ERIVE THE QUADRATIC FORMULA Use the quadratic formula to solve equations in standard form. If the equation is not in standard form, use algebra to put the equation into standard form

U SING THE Q UADRATIC E QUATION What are the solutions to the following equations? 2x 2 – x = 4 x 2 + 4x = -4 x 2 + 4x – 3= 0 5x 2 – 2x = 2

E XAMPLE You sell wrapping paper as a charity fundraiser. The equation p = -6x x models the total profit p as a function of the price x per roll of paper. What is the smallest amount in dollars you can charge per roll of wrapping paper to make a profit of $1500?

Solutions to Quadratics Two real solutions, one real solution, or not real solutions Discriminant : the value of b 2 – 4ac This tells you how many real solutions an equation has. If b 2 – 4ac > 0 there are two real solutions. What does this graph look like? If b 2 – 4ac = 0 there is one real solution. What does this graph look like? If b 2 – 4ac < 0 there is no real solution. What would this graph look like?

What is the number of real solutions to the equations: -2x 2 – 3x + 5 = 0? 2x 2 – 3x + 7 x 2 = 6x + 5 -x x = 49

E XAMPLE A rocket is launched from the ground with an initial vertical velocity of 150 ft/s. The function h = -16t t models the height in feet of the rocket at time t in seconds. Will the rocket reach a height of 300 ft? Explain your answer.