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Solving Quadratic Equation by Graphing and Factoring

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Section 6.2& 6.3

CCSS: A.REI.4b

- 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.

- 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.

- 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?

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

Example f(x) = 4x2 - 3

Quadratic term 4x2

Linear term 0

Constant term -3

Now you try this problem.

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

quadratic term

linear term

constant term

5x2

-2x

3

The number of real solutions is at most two.

No solutions

One solution

Two solutions

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

Solutions are -2 and 2.

Now you try this problem.

f(x) = 2x - x2

Solutions are 0 and 2.

- 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.

x

y

0

0

1

-3

2

-4

3

-3

4

0

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

x

y

-2

-1

1

3

4

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

Roots

Vertex

Axis of Symmetry

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

- 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

Make one side zero.

Then factor then set each factor to zero

Solve

Solve

Solve

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

Use -6 and 1 to break up the middle term

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

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

Then foil the factors,

x2 - (r1 + r2)x+(r1· r2)=0

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

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