Modeling with quadratic equations

About this lesson

Algebra I › Quadratic equations

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Modeling with quadratic equations

Warm-up

Three quick questions from earlier lessons.

Discover

A square has an area of 49 cm². Which equation gives its side s?

  1. s² = 49
  2. 4s = 49

Key idea

Write the equation from the situation, solve it, then check the context.

Name the unknown, translate the words into an equation (often a product: area = length × width, or a formula such as h=−5t2+vth = -5t^2 + vt), solve it, and keep only the solutions that make sense: lengths and times are not negative, and the launch time is not the landing time.

Worked example

A rectangle is 3 m longer than wide, with area 40 m².

w(w+3)=40w(w + 3) = 40
Equation.
w2+3w−40=0w^2 + 3w - 40 = 0, (w+8)(w−5)=0(w + 8)(w - 5) = 0
Solve.
w=5w = 5 m (reject −8-8)
Context.

Your turn, guided

Your turn: a square of area 81 m². Its side, in meters?

99
s² = 81, s > 0.

Practice ★

  1. The product of two consecutive whole numbers is 240. Which equation models this, with x the smaller number?
    1. x(x + 1) = 240
    2. x + (x + 1) = 240
    3. x² + 1 = 240
  2. The product of two consecutive whole numbers is 210. Which equation models this, with x the smaller number?
    1. x(x + 1) = 210
    2. x + (x + 1) = 210
    3. x² + 1 = 210
  1. A rectangular garden is 6 m longer than it is wide. Its area is 216 m². Write and solve an equation for the width w. What is the width, in meters?

Watch out!

Practice ★★

  1. A ball is kicked upward. Its height after t seconds is h = −5t² + 35t meters. After how many seconds does it land?
  2. A ball is kicked upward. Its height after t seconds is h = −5t² + 30t meters. After how many seconds does it land?
  1. Boss problem. A photo 22 cm by 14 cm gets a frame of the same width x cm on every side. The framed photo covers 660 cm².

Exit ticket

Three questions to check you have got it.

  1. A rectangular garden is 6 m longer than it is wide. Its area is 216 m². Write and solve an equation for the width w. What is the width, in meters?
  2. A rectangular garden is 3 m longer than it is wide. Its area is 88 m². Write and solve an equation for the width w. What is the width, in meters?
  3. A rectangular garden is 7 m longer than it is wide. Its area is 144 m². Write and solve an equation for the width w. What is the width, in meters?

Summary card

Modeling with quadratic equations

Translate, solve, then keep the solutions that fit the situation.

Watch out: a negative length or the start time is not the answer.

End of the lesson

Next lesson: Rational and irrational numbers; repeating decimals Back to the chapter

Answers: Modeling with quadratic equations

A

  1. x(x + 1) = 240
  2. x(x + 1) = 210

B

  1. 1212

C

  1. 77
  2. 66

D

  1. 352352

E

  1. 1212
  2. 88
  3. 99

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