solving quadratic equation by graphing and factoring section 6.2& 6.3 ccss: a.rei.4b
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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. 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
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