Download 8a: Solving Quadratic Equations Using Square Roots

8a: Solving Quadratic Equations Using Square Roots
Algebra 1 CCSS A.REI.4.b
Essential Question:
How do you solve a quadratic equation using square roots?
Vocabulary

Quadratic
Equation
An equation whose highest degree is 2 or highest exponent is 2; usually written in the
form of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Solution/
Roots/Zeros
The solution of a quadratic equation where its parabola intersects the x-axis
Skills Check – Simplify each expression.
1) √36
What are possible
answers when
simplifying a square
root?
2) −√81
49
3) √100
16
4) −√121
You can solve equations with squared variables (i.e. 𝑥 2 ) by using its inverse
operation, square root √𝑥 2 = ±𝑥. Notice that the answer to a square root is the
negative and positive of a number.
√16 = ±4
√64 = ±8
√100 = ±10
Why is the answer to a square root the positive and negative of a number?
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How do I simplify a
radical expression
when it is not a
perfect square?
A radical expression can be written as two separate radical expressions that are
multiplied with each other.
√𝑎𝑏 = √𝑎 ∙ √𝑏
To simplify a radical expression, remove perfect square factors of expression.
√54 = √9 ∙ √6
= ±3 ∙ √6
= ±3√6
Practice – Simplify each radical expression.
1) √50
How do I solve
quadratic equations
using square roots?
2) √48
3) −5√300
4) −3√18
Solving a quadratic equation using square roots:
1. Isolate the squared variable using inverse operations.
2. Square root both sides of the equation.
3. Remember to include the positive and negative of your final answer.
Practice – Solve each quadratic equation using square roots.
a) 𝑡 2 − 25 = 0
b) 2𝑔2 + 32 = 0
c) 2𝑥 2 − 98 = 0
d) 3𝑛2 + 12 = 12
When would a
quadratic equation
have no solution?
Try this: 𝑚2 + 25 = 0
Explain why you could not determine the solution.
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Practice – Solve each equation by finding the square roots. If the equation has
no solution, write no solution.
e) 𝑚2 − 225 = 0
f) 𝑐 2 + 25 = 25
g) 𝑥 2 − 9 = −16
h) 27 − 𝑦 2 = 0
i) Michael’s work is shown below. Explain the error he made.
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8b: Solving Quadratic Equations by Factoring
Algebra 1 CCSS A.REI.4.b
Essential Question:
How do I solve quadratic equations by factoring?
Vocabulary:

Zero Product For every real number 𝑎 and 𝑏, if 𝑎𝑏 = 0, then 𝑎 = 0 or 𝑏 = 0.
Property
In other words, when multiplying two numbers, in order to get a product of 0, one of
the two numbers have to be 0.
Skills Check – Factor each quadratic expression.
a) 𝑥 2 + 5𝑥 + 4
How do I solve
equations using the
Zero Product
Property?
b) 8𝑥 2 + 8𝑥 − 6
1. Separate and set each binomial parenthesis equal to 0.
2. Solve each equation.
3. The answers represent the roots to the quadratic equation.
Practice – Determine the roots for each equation.
c) (𝑥 + 5)(2𝑥 − 6) = 0
d) −3𝑛(2𝑛 − 10) = 0
How do I solve
quadratic equations
by factoring?
You can also use the Zero Product Property to solve equations in the form of
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 if the equation can be factored.
1.
2.
3.
4.
5.
Make sure the equation is in standard form (𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0)
Factor the quadratic equation.
Use the Zero Product Property to separate and set each factor equal to 0.
Solve each equation.
The answers represent the roots of the quadratic equation.
Practice – Determine the solution for each quadratic equation.
e) 𝑥 2 + 6𝑥 + 8 = 0
f) 2𝑥 2 − 5𝑥 = 88
g) 2𝑥 2 − 9𝑥 = 0
h) 4𝑦 2 = 25
i) The sides of a small square are increased by 6 centimeters. The area of the new
square is 81 square centimeters. Find the length of a side of the original square.
j) You are building a rectangular pool. You want the area to be 120 ft. You also want
the length of the pool to be 8 feet more than twice its width. Find the dimensions of
the pool.
Summary: In your own words, describe the Zero Product Property.
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8c: Completing the Square
Algebra 1 CCSS A.REI.4.b
Essential Question:
Why do we complete the square to solve quadratic equations?
Skills Check – Determine the solutions to the quadratic equation.
a) 𝑥 2 + 10𝑥 + 25 = 121
How can you create
a perfect square
trinomial?
You can make a quadratic equation a perfect square trinomial by completing the
square.
To make a perfect square trinomial, take half the value of b and square it:
𝑏 2
( )
2
Practice – Find the value that would complete the square.
b) 𝑥 2 − 6𝑥
c) 𝑦 2 + 8𝑦
d) 𝑥 2 + 9𝑥
How do I find the
roots of a quadratic
equation by
completing the
square?
1. Isolate the 𝑐 term if necessary.
𝑏 2
2. Find the value that would complete the square: (2)
3. Factor the perfect square trinomial.
4. Solve using square roots.
Practice – Determine the solutions by completing the square.
e) 𝑟 2 + 8𝑟 = 48
f) 𝑞 2 + 22𝑞 = −85
g) 𝑥 2 − 6𝑥 − 7 = 0
h) 𝑥 2 + 8𝑥 + 12 = 0
i) 𝑥 2 + 10𝑥 = −17
j) 𝑥 2 − 2𝑥 − 5 = 0
8d: The Quadratic Formula
Algebra 1 CCSS A.REI.4.b
Essential Question:
How can you find the solution to a quadratic equation that can’t be factored?
Vocabulary:

Quadratic
Formula
The quadratic formula is another method of finding the solution to a quadratic
equation:
Quadratic Formula:
How do I use the
quadratic formula?
1. Label the 𝑎, 𝑏, and 𝑐.
2. Substitute each value into the quadratic formula.
3. Simplify.
Practice – Use the Quadratic Formula to determine the solution to each
quadratic function.
1) 𝑥 2 − 2𝑥 − 8 = 0
2) 𝑥 2 + 6 = 5𝑥
3) 2𝑥 2 + 4𝑥 − 7 = 0
5) 7𝑥 2 − 2𝑥 − 8 = 0
4) −3𝑥 2 + 5𝑥 − 2 = 0
6) Which equation does
9±√(−9)2 −4(5)(−7)
2(5)
to?
give the solutions
8e: The Discriminant
Algebra 1 CCSS A.REI.4.b
Essential Question:
What is the discriminant and what information does it provide?
Vocabulary:

Discriminant
The expression under the radical in the quadratic formula; the discriminant tells you
the number of solutions to a quadratic equation
A quadratic equation can have one solution, two solutions, or no solutions.
How can I
determine the
number of solution/s 1) Substitute values of 𝑎. 𝑏, and 𝑐 into the discriminant: 𝑏 2 − 4𝑎𝑐
a quadratic
equation will have?
 If 𝑏 2 − 4𝑎𝑐 > 0, then there are TWO solutions
 If 𝑏 2 − 4𝑎𝑐 < 0, then there are NO solutions
 If 𝑏 2 − 4𝑎𝑐 = 0, then there is ONE solution
Practice – Determine how many times each quadratic equation intersects the xaxis.
1) 0 = 4𝑥 2 − 12𝑥 = 9
2) 3𝑥 2 − 5𝑥 = 1
3) 3𝑥 2 − 4𝑥 = 7
4) 𝑥 2 + 𝑥 = 0
5) Ken claims that the discriminant of 2𝑥 2 + 5𝑥 − 1 = 0 is 17. What error did he
make?
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What is the correct discriminant?