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

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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


Solve

Solve


Solve1

Solve


Solve2

Solve


Solve3

Solve


Solving quadratic equation by graphing and factoring

Solve


Solve4

Solve

Solve


Solving quadratic equation by graphing and factoring

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


Solve7

Solve


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

Equationx2 - (-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

Equationx2 -(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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