Solving quadratic equation by graphing and factoring
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Solving Quadratic Equation by Graphing and Factoring. Section 6.2& 6.3 CCSS: A.REI.4b. Mathematical Practices:. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others.  

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Solving quadratic equation by graphing and factoring
Solving Quadratic Equation by Graphing and Factoring

Section 6.2& 6.3

CCSS: A.REI.4b


Mathematical practices
Mathematical Practices:

  • 1. Make sense of problems and persevere in solving them.

  • 2. Reason abstractly and quantitatively.

  • 3. Construct viable arguments and critique the reasoning of others.  

  • 4. Model with mathematics.

  • 5. Use appropriate tools strategically.

  • 6. Attend to precision.

  • 7. Look for and make use of structure.

  • 8. Look for and express regularity in repeated reasoning.


Ccss a rei 4b
CCSS: A.REI.4b

  • SOLVE quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. RECOGNIZE when the quadratic formula gives complex solutions and write them as a ± bifor real numbers a and b.


Essential question
Essential Question:

  • How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph & how do I solve quadratic functions by factoring?


Quadratic equation
Quadratic Equation

y = ax2 + bx + c

ax2__ is the quadratic term.

bx--- is the linear term.

c-- is the constant term.

The highest exponent is two; therefore, the degree is two.


Identifying terms
Identifying Terms

Example f(x)=5x2-7x+1

Quadratic term 5x2

Linear term -7x

Constant term 1


Identifying terms1
Identifying Terms

Example f(x) = 4x2 - 3

Quadratic term 4x2

Linear term 0

Constant term -3


Identifying terms2
Identifying Terms

Now you try this problem.

f(x) = 5x2 - 2x + 3

quadratic term

linear term

constant term

5x2

-2x

3


Quadratic solutions
Quadratic Solutions

The number of real solutions is at most two.

No solutions

One solution

Two solutions


Solving equations
Solving Equations

When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.

These values are also referred to as solutions, zeros, or roots.


Identifying solutions
Identifying Solutions

Example f(x) = x2 - 4

Solutions are -2 and 2.


Identifying solutions1
Identifying Solutions

Now you try this problem.

f(x) = 2x - x2

Solutions are 0 and 2.


Graphing quadratic equations
Graphing Quadratic Equations

  • The graph of a quadratic equation is a parabola.

  • The roots or zeros are the x-intercepts.

  • The vertex is the maximum or minimum point.

  • All parabolas have an axis of symmetry.


Graphing quadratic equations1

x

y

0

0

1

-3

2

-4

3

-3

4

0

Graphing Quadratic Equations

One method of graphing uses a table with arbitrary

x-values.

Graph y = x2 - 4x

Roots 0 and 4 , Vertex (2, -4) ,

Axis of Symmetry x = 2


Graphing quadratic equations2

x

y

-2

-1

1

3

4

Graphing Quadratic Equations

Try this problem y = x2 - 2x - 8.

Roots

Vertex

Axis of Symmetry


Graphing quadratic equations3
Graphing Quadratic Equations

The graphing calculator is also a helpful tool for graphing quadratic equations.


Roots or zeros of the quadratic equation
Roots or Zeros of the Quadratic Equation

  • The Roots or Zeros of the Quadratic Equation are the points where the graph hits the x axis.

  • The zeros of the functions are the input that make the equation equal zero.

    Roots are 4,-3


To solve a quadratic equation
To solve a Quadratic Equation

Make one side zero.

Then factor then set each factor to zero







Solve4

Solve

Solve


Solve

Multiply the ends together and find what adds to the coefficient of the middle term


Solve5
Solve

Use -6 and 1 to break up the middle term


Solve6
Solve

Use group factoring to factor, first two terms and then the last two terms



How to write a quadratic equation with roots
How to write a quadratic equation with roots

Given r1,r2 the equation is (x - r1)(x - r2)=0

Then foil the factors,

x2 - (r1 + r2)x+(r1· r2)=0


How to write a quadratic equation with roots1
How to write a quadratic equation with roots

Given r1,r2 the equation is (x - r1)(x - r2)=0

Then foil the factors,

x2 - (r1 + r2)x+(r1· r2)=0

Roots are -2, 5

Equation x2 - (-2+5)x+(-2)(5)=0

x2 - 3x -10 = 0


How to write a quadratic equation with roots2
How to write a quadratic equation with roots

Roots are ¼, 8

Equation x2 -(¼+8)x+(¼)(8)=0

x2 -(33/4)x + 2 = 0

Must get rid of the fraction, multiply by the common dominator. 4

4x2 - 33x + 8 = 0


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