options: Work with equal groups of objects to gain foundations for multiplication.

Rewrite rational expressions.

Interpret the structure of expressions.

Analyze patterns and relationships.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Perform arithmetic operations on polynomials.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Understand solving equations as a process of reasoning and explain the reasoning.

Use properties of operations to generate equivalent expressions.

Create equations that describe numbers or relationships

Represent and analyze quantitative relationships between dependent and independent variables.

Reason about and solve one-variable equations and inequalities.

Work with addition and subtraction equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand the relationship between zeros and factors of polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Add and subtract within 20.

Represent and solve problems involving multiplication and division.

Use the four operations with whole numbers to solve problems.

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving addition and subtraction.

Understand properties of multiplication and the relationship between multiplication and division.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve systems of equations.

Multiply and divide within 100.

Solve equations and inequalities in one variable.

Work with radicals and integer exponents.

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Understand the connections between proportional relationships, lines, and linear equations.
problem_activity: Task
In ''Building an explicit quadratic function'' a particular quadratic function was rewritten by completing the square. The quadratic function used was $q(x) = 2x^2 + 4x - 16$ and this function was rewritten as $$ q(x) = 2\left(x+1\right)^2 - 18 = 2\left((x+1)^2 - 9\right). $$ Some of the advantages to this form are that the $x$-coordinate of the vertex of the graph of $q$ can be found more easily and the two roots of $q$ can also be found readily. The right hand side of this equation can be seen as a horizontal translation by $-1$, then squaring, then a vertical translation by $-9$, and finally a multiplicative scaling by $2$. The goal of this task is first to employ the same technique on a general quadratic function and then derive the quadratic formula. To assist in this process, we first rewrite the equation above: $$ \frac{q(x)}{2} = \left(x+1\right)^2 - 9. $$
Let $f$ be a quadratic function, so we have $$ f(x) = ax^2 + bx + c. $$ Here $a,b,c$ are real numbers and we assume that $a$ is non-zero.
Following the lead of our example problem, we begin by dividing out the leading coefficient:
Multiplying $f(x)$ by $\frac{1}{a}$ gives $$ \frac{f(x)}{a} = x^2 + \frac{b}{a}x + \frac{c}{a}. $$ We wish to write this as the square of a linear function $x + d$ plus a constant. Find numbers $d$ and $k$ so that $\frac{f(x)}{a} = (x+d)^2 + k$.
Using the values for $d$ and $k$ found in part (b), rewrite $f(x) = ax^2 + bx + c$ as $$ \frac{f(x)}{a} = \left(x+d\right)^2 + k. $$ Why do the expressions $f(x)$ and $\frac{f(x)}{a}$ have the same roots?
Explain how to deduce the quadratic formula for the roots of $f$ from part (c).


correct_option_index: 21

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