Formulating Quadratic Models

Learn the concept

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 45 · 1 point

Worked example

A 24-foot by 40-foot garden is enlarged by adding x feet to its width and 2x feet to its length. The new area cannot exceed 1,600 square feet. Write an inequality.

1. Let x be the nonnegative number of feet added to the width.
2. The new width is $24+x$ and the new length is $40+2x$.
3. Multiply the dimensions: $A=(24+x)(40+2x)$.
4. Use ≤ because the area cannot exceed the stated limit.

$(24+x)(40+2x)\le 1600$, with $x\ge 0$.

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Try it yourself

TEKSForge practice · unscored

1. 1. The product of two consecutive positive odd integers is 195. Let n be the smaller integer. Write an equation.

   Show answer and reasoning

   $n(n+2)=195$, with n a positive odd integer.

   The integers differ by 2, and product means multiplication. The model is satisfied by 13 and 15.

2. 2. A 9-meter by 16-meter rectangle gains 2t meters in width and t meters in length. Write a model requiring area greater than 220 square meters.

   Show answer and reasoning

   $(9+2t)(16+t)>220$, with $t\ge 0$.

   Each change stays with its stated dimension. “Greater than” uses a strict inequality.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 23 · 1 point

The product of 2 consecutive odd integers is 483. If $x$ represents the smaller integer, which equation can be used to find both integers?

1. $x(x + 2) = 483$
2. $x(x + 1) = 483$
3. $x(x + 3) = 483$
4. $(x + 1)(x + 2) = 483$

Show STAAR answer and explanation

$x(x + 2) = 483$

Consecutive odd integers differ by 2, so the larger integer is $x+2$ and their product is $x(x+2)=483$. The integers are 21 and 23.

$x(x + 1) = 483$: $x+1$ is the next integer, which is even when x is odd.

$x(x + 3) = 483$: $x+3$ is even when x is odd; consecutive odd integers are 2 apart.

$(x + 1)(x + 2) = 483$: Neither factor is x, the smaller integer, and these factors differ by only 1.

Ask the tutor about this question

Archived released STAAR Algebra II EOC · 2013 · Item 45 · 1 point

A rectangular parking lot measures 150 feet by 230 feet. The owner of the parking lot will expand the lot's size by adding $x$ feet and $2x$ feet to its dimensions, as shown. If the total area of the parking lot cannot exceed 40,000 square feet, which inequality can be used to find all possible values of $x$?

1. $(230 + 2x)(150 + x) \le 40{,}000$
2. $(230 + x)(150 + 2x) \le 40{,}000$
3. $(230 + 2x)(150 + x) \ge 40{,}000$
4. $(230 + x)(150 + 2x) \ge 40{,}000$

Show STAAR answer and explanation

$(230 + 2x)(150 + x) \le 40{,}000$

The figure adds $2x$ to the 230-foot length and $x$ to the 150-foot width, so the new area is $(230+2x)(150+x)$. “Cannot exceed” means less than or equal to.

$(230 + x)(150 + 2x) \le 40{,}000$: The figure adds 2x to the 230-foot side and x to the 150-foot side, not the reverse.

$(230 + 2x)(150 + x) \ge 40{,}000$: “Cannot exceed” means $\le$, not $\ge$.

$(230 + x)(150 + 2x) \ge 40{,}000$: Both the side lengths and the inequality symbol are reversed.

Ask the tutor about this question