Objective:Solving Quadratic Equations by Finding Square Roots

This lesson comes from chapter 9.1 from your

textbook, page 503

Quadratic Equations

Standard form: ax2 + bx + c = 0

Find the square root of numbers𝟖𝟐=¿𝟔𝟒√𝟔𝟒=¿𝟖

−√𝟔𝟒=¿ −𝟖

√𝟎=¿𝟎√−𝟒=¿𝟐 𝒊𝒐𝒓 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

Find the square root of numbers

√𝟓𝟎−𝟏𝟒=¿√𝟑𝟔=𝟔

−√𝟐𝟐+𝟓=¿−√𝟗=−𝟑

1.

2.

Find the square root of numbers

√𝟒 (𝟐 )+𝟐𝟑=¿√𝟏𝟔=𝟒

√𝟑¿¿

1.

2. √𝟐𝟓=𝟓

Find the square root of numbers𝒙𝟐=𝟗

𝒙=±𝟑

√𝒙𝟐=√𝟗

1.

Key ConceptsWhen x2 = d

If d > 0, then x2 = d has two solutions

example:

If d = 0, then x2 = d has one solution

example:

If d < 0, then x2 = d has no real solution

example:

𝒙𝟐=𝟏𝟎𝟎

𝒙𝟐=−𝟗

𝒙𝟐=𝟎

Find the square root of numbers1. 2. 𝒙𝟐=𝟑𝟔

𝒙=±𝟔

√𝒙𝟐=√𝟑𝟔𝒃𝟐=𝟎

𝒃=𝟎

√𝒃𝟐=√𝟎

Find the square root of numbers1. 2. 𝒚𝟐+𝟑=𝟐𝟖

𝒚=±𝟓

√𝒚𝟐=√𝟐𝟓

−𝟑−𝟑𝒚𝟐=𝟐𝟓

𝒙𝟐−𝟒=𝟒𝟓

𝒙=±𝟕

√𝒙𝟐=√𝟒𝟗

+𝟒+𝟒𝒙𝟐=𝟒𝟗

Find the square root of numbers1. 2. 𝟐𝒎𝟐=𝟑𝟐

𝒎=±𝟒

√𝒎𝟐=√𝟏𝟔

𝟐𝟐𝒎𝟐=𝟏𝟔

𝟐𝒌𝟐−𝟖=𝟗𝟎

√𝒌𝟐=√𝟒𝟗

𝟐𝟐𝒌𝟐=𝟒𝟗

+𝟖+𝟖𝟐𝒌𝟐=𝟗𝟖

𝒌=±𝟕

Find the square root of numbers1. 2. 𝟐𝒏𝟐+𝟑=𝟓𝟑

𝒏=±𝟓√𝒏𝟐=√𝟐𝟓

𝟐𝟐𝒏𝟐=𝟐𝟓

−𝟑−𝟑𝟐𝒏𝟐=𝟓𝟎

𝟒 𝒙𝟐+𝟓=𝟔𝟗

𝒙=±𝟒

√𝒙𝟐=√𝟏𝟔

𝟒𝟒𝒙𝟐=𝟏𝟔

−𝟓−𝟓𝟒 𝒙𝟐=𝟔𝟒

Real Life: Equation of a falling object

When an object is dropped, the speed with which it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model.

sth 216h is the ending height in feet above the groundt is the number of seconds the object has been fallings is the starting height from which the object was dropped

Application

Sarah is going to drop a water balloon from a height of 144 feet. To the nearest tenth of a second, about how long will it take for the balloon to hit the ground? Assume there is no air resistance.

The question asks to find the time it takes for the container to hit the ground.

Initial height (s) = 144 feet

Height when its ground (h) = 0 feet

Time it takes to hit ground (t) = unknown

sth 216

Substitutesth 216

𝟎=−𝟏𝟔𝒕𝟐+𝟏𝟒𝟒

3 sec.

√𝟗=√𝒕𝟐

−𝟏𝟔−𝟏𝟔

−𝟏𝟒𝟒−𝟏𝟒𝟒−𝟏𝟒𝟒=−𝟏𝟔𝒕𝟐

𝟗=𝒕𝟐

Substitutesth 216

𝟎=−𝟏𝟔𝒕𝟐+𝟐𝟖𝟖

4.24 sec.

√𝟏𝟖=√𝒕𝟐

−𝟏𝟔−𝟏𝟔

−𝟐𝟖𝟖−𝟐𝟖𝟖−𝟐𝟖𝟖=−𝟏𝟔𝒕𝟐

𝟏𝟖=𝒕𝟐

Find the square root of numbers1. 2. 𝟕𝒉𝟐−𝟒=𝟑

𝒉=±𝟏√𝒉𝟐=√𝟏

𝟕𝟕𝒉𝟐=𝟏

+𝟒+𝟒𝟕𝒉𝟐=𝟕

𝟒𝒑𝟐+𝟕=𝟒𝟑−𝟕−𝟕𝟒𝒑𝟐=𝟑𝟔𝟒𝟒𝒑𝟐=𝟗√𝒑𝟐=√𝟗

𝒑=±𝟑

Application

An engineering student is in an “egg dropping contest.” The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking the egg. To the nearest tenth of a second, about how long will it take for the egg’s container to hit the ground? Assume there is no air resistance.

The question asks to find the time it takes for the container to hit the ground.

Initial height (s) = 32 feet

Height when its ground (h) = 0 feet

Time it takes to hit ground (t) = unknown

sth 216

Substitutesth 216

𝟎=−𝟏𝟔𝒕𝟐+𝟑𝟐

Approximately 1.4 sec.

√𝟐=√𝒕𝟐

−𝟏𝟔−𝟏𝟔

−𝟑𝟐−𝟑𝟐−𝟑𝟐=−𝟏𝟔𝒕𝟐

𝟐=𝒕𝟐

Evaluate a Radical Expression

√𝒃𝟐−𝟒𝒂𝒄=√(−𝟐)𝟐−𝟒(𝟏)(−𝟑)=√𝟒−𝟒(−𝟑)

√𝟒+𝟏𝟐=√𝟏𝟔=𝟒

Perfect Squares: Numbers whose square roots are integers or quotients of integers.

1316912144

1112110100

981864749

636525416

392411

What is a square root?

If a number square (b2) = another number (a), then b is the square root of a.

Example: If 32 = 9, then 3 is the square root of 9

Quadratic Equations

Standard form: ax2 + bx + c = 0a is the leading coefficient and cannot be equal to zero.If the value of b were equal to zero, the equation becomes ax2 + c = 0.We can solve equations is this form by taking the square root of both sides.

Some basics…

All positive numbers have two square rootsThe 1st is a positive square root, or principal square root.The 2nd is a negative square rootSquare roots are written with a radical symbol You can show both square roots by using the “plus-minus” symbol ±