Quadratic Formula

Grade 9•45 minutes•A.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A.REI.B.4.a Use the method of completing the square to transform any quadratic equation in 𝑥 into an equation of the form (𝑥 – 𝑝)² = 𝑞 that has the same solutions. Derive the quadratic formula from this form. A.REI.B.4.b Solve quadratic equations by inspection (e.g., for 𝑥² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏. A.REI.C.6 Solve systems of linear equations algebraically (include using the elimination method) and graphically, focusing on pairs of linear equations in two variables. A.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A.REI.D.11 Explain why the 𝑥-coordinates of the points where the graphs of the equations 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥) = 𝑔(𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A.SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients. F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). N.CN.A.2 Use the relation 𝑖² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

Learning Objective

I can use the quadratic formula to find the real and complex solutions of quadratic equations in standard form.

Practice Questions

This lesson includes 12 practice questions to reinforce learning.

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1. What is the quadratic formula? Explain what each variable (a, b, c) represents in the context of a quadratic equation in standard form.

2. Solve the following quadratic equation using the quadratic formula: x² + 5x + 6 = 0. Show your steps.

3. A small business is analyzing its profit margin, which can be modeled by the equation -0.5x² + 4x - 3 = 0, where x represents the number of units sold. Use the quadratic formula to determine the number of units they need to sell to break even (i.e., when the profit margin is zero).

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How To Solve Quadratic Equations Using The Quadratic Formula

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