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.   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

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 ± bi for real numbers a and b.

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

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.

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

Quadratic term 5x2

Linear term -7x Constant term 1

Identifying Terms

Example f(x) = 4x2 - 3

Quadratic term 4x2

Linear term 0Constant term -3

Identifying Terms

Now you try this problem.

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

quadratic term linear term constant term

Identifying Terms

5x2

-2x

3

The number of real solutions is at most two.

Quadratic Solutions

No solutions

6

4

2

-2

5

f x = x2-2 x +5

6

4

2

-2

5

2

-2

-4

-5 5

One solution Two solutions

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.

Example f(x) = x2 - 4

Identifying Solutions

4

2

-2

-4

-5 5

Solutions are -2 and 2.

Now you try this problem.

f(x) = 2x - x2

Solutions are 0 and 2.

Identifying Solutions

4

2

-2

-4

5

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 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 Equations

x y0 01 -32 -43 -34 0

4

2

-2

-4

5

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

RootsVertexAxis of Symmetry

Graphing Quadratic Equations

x y-2-1134

4

2

-2

-4

5

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

Graphing Quadratic Equations

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,-3 034 xx

To solve a Quadratic Equation

Make one side zero.

Then factor then set each factor to zero

05

5

2

2

xx

xx

5;0

05;0

05

xx

xx

xx

Solve

xx 3282

Solve

0283

328

2

2

xx

xx

047

0283

328

2

2

xx

xx

xx

Solve

04;07

047

0283

328

2

2

xx

xx

xx

xx

Solve

4 ;7

04;07

047

0283

328

2

2

xx

xx

xx

xx

xx

Solve

Solve

0253

253

2

2

xx

xx

Solve

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

0253

253

2

2

xx

xx

5)1()6(

6)1)(6(

6)2(3

Solve

Use -6 and 1 to break up the middle term

0253

253

2

2

xx

xx

02163 2 xxx

Solve

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

02163 2 xxx

0132

02123

xx

xxx

Solve

Solve

02163 2 xxx

3

1 ;2

13 ;2

0132

02123

xx

xx

xx

xxx

0253

253

2

2

xx

xx

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 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 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