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MAT 105 · Unit 1 ALEKS · objective 1.2

Modeling with Quadratic Equations

Translate area, motion, and number relationships into quadratics and interpret valid roots.

Standalone concept package
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Build the concept from the beginning

Starting point

No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.

Learning target

Translate area, motion, and number relationships into quadratics and interpret valid roots.

Evidence of mastery

Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.

Essential terms

equation
A statement that two mathematical expressions have the same value.
solution
A value that makes an equation true when substituted for the variable.
factor
A quantity multiplied by another quantity to form a product.
root
A solution of an equation; for a polynomial, it makes the polynomial equal zero.
Look first, then name the mathematics

Build the visual meaning

A situation becomes a quadratic equation, branches to algebraic roots, and passes each root through a units-and-domain check.
How to read this visual: A model-to-equation-to-roots map separates algebraic solutions from contextually valid answers.
  • Define variable
  • Build quadratic
  • Solve roots
  • Check domain
  • Interpret units
Read the complete visual relationship as text
  • Define variable
  • Build quadratic
  • Solve roots
  • Check domain
  • Interpret units
Definition in plain language

A quadratic model arises when an unknown is multiplied by itself or another expression containing the unknown, or when change includes a squared term.

Why this matters

Algebra may produce two roots, but units, time, length, and stated conditions determine which roots make sense in the situation.

Reasoning path

See the method as a sequence

Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.

Define the unknown and its domain.
Translate the relationship into a quadratic equation.
Solve with an appropriate method.
Check each root and reject only those excluded by the context.

The same method in words

  1. Define the unknown and its domain.
  2. Translate the relationship into a quadratic equation.
  3. Solve with an appropriate method.
  4. Check each root and reject only those excluded by the context.
Interactive decision model

Build the reasoning, one decision at a time

Model a rectangle with width w, length w + 4, and area 96.

Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.

The reasoning path changes as each decision is selected.

Model before practice

Three fully explained examples

Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.

Rectangle area

Name the goalLet width be w and length be w + 3. Area is 40.
Choose the next moveWrite w(w + 3) = 40.
Carry out the mathematicsw² + 3w − 40 = 0 = (w + 8)(w − 5).
Check and interpretRoots are −8 and 5; width must be positive, so w = 5 m.

Projectile landing

Name the goalHeight is h = −5t² + 20t. Ground means h = 0.
Choose the next moveWrite −5t² + 20t = 0 and factor.
Carry out the mathematics−5t(t − 4) = 0 gives t = 0 or 4.
Check and interprett = 0 is launch; the ball returns to ground at 4 seconds.

Consecutive integers

Name the goalLet n be the smaller positive integer; the next is n + 1. Product is 72.
Choose the next moveWrite n(n + 1) = 72.
Carry out the mathematicsn² + n − 72 = 0 = (n + 9)(n − 8).
Check and interpretThe positive pair is 8 and 9; −9 is outside the stated domain.

Practice with decreasing support

Guided practice

Practice immediately after the model

Predict first. Then use the visual relationship and reasoned feedback to check two decisions.

1. A rectangle has area 48 m² and length 2 m more than width. What is the width?

2. A ball's height is h = −5t² + 20t. When does it return to the ground after launch?

Not completed yet.
Partially completed problem

Finish the missing step

A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.

1. Two consecutive positive integers have product 72. What is the smaller?

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Independent practice

Six problems for repetition

Solve all six. Use the specific feedback to revise a method or sign error.

1. A square has area 121 cm². What is its side length?

2. A garden is 3 m longer than wide and has area 40 m². What is its width?

3. A projectile has h = −16t² + 64t + 80. When does it hit the ground?

4. The product of a number and 4 more than the number is 96. What positive number works?

5. A rectangle's width is 5 cm less than its length and area is 84 cm². What is the length?

6. A ball's height is h = −t² + 6t + 7. When is h = 0 after launch?

Not completed yet.

Mastery and transfer

Mastery check

Demonstrate understanding

Attempt all five. At least four correct answers are required for mastery.

1. A right triangle has legs x and x + 7 and hypotenuse 13. What is the shorter leg?

2. Revenue is R = −2x² + 40x. At what positive x is revenue zero?

3. A rectangular poster has area 180 in² and length 3 in more than width. What is the width?

4. The square of a number is 10 more than three times the number. What are the solutions?

5. A path surrounds a 10 m by 6 m garden. The total area is 192 m². What is the path width?

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Unfamiliar transfer

Explain, compare, and revise

Create a quadratic model with two algebraic roots and explain which roots are meaningful in context.

Write a first attempt before opening the self-check.

The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.