This document provides examples of using the discriminant of a quadratic equation to determine the number of solutions. It defines the discriminant as b^2 - 4ac in the quadratic formula ax^2 + bx + c = 0. Several examples show calculating the discriminant of example equations and using whether it is positive, zero, or negative to determine if there are two, one, or no real solutions. The last example is a word problem about the height of a water fountain that is solved using the discriminant to find where the fountain reaches a certain height.
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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Question 1 of 502.0 PointsSimplify the complex rational expres.docxmakdul
Question 1 of 50
2.0 Points
Simplify the complex rational expression.
A.
B.
C.
D.
Question 2 of 50
2.0 Points
Solve the quadratic equation by the square root property. (2x + 5) 2 = 49
A. {-6, 1}
B. {0, 1}
C. {-27, 27}
D. {1, 6}
Question 3 of 50
2.0 Points
Solve the linear equation.
A.
B.
C.
D.
Question 4 of 50
2.0 Points
Write the number in scientific notation.
0.000779
A.
7.79 x 10 -4
B.
7.79 x 10 4
C.
7.79 x 10 -3
D.
7.79 x 10 -5
Question 5 of 50
2.0 Points
Graph the equation in the rectangular coordinate system.
3y = 15
A.
B.
C.
D.
Question 6 of 50
2.0 Points
Along with incomes, people's charitable contributions have steadily increased over the past few years. The table below shows the average deduction for charitable contributions reported on individual income tax returns for the period 1993 to 1998. Find the average annual increase between 1995 and 1997.
A. $270 per year
B. $280 per year
C. $335 per year
D. $540 per year
Question 7 of 50
2.0 Points
Find the domain of the function.
A.
(-∞, 6) (6, ∞)
B.
C.
D.
(-∞, 6]
Question 8 of 50
2.0 Points
Graph the line whose equation is given.
A.
B.
C.
D.
Question 9 of 50
2.0 Points
Find the zeros of the polynomial function.
f(x) = x 3 + 5x 2 – 4x – 20
A. x = –5, x = 4
B. x = –2, x = 2
C. x = –5, x = –2, x = 2
D. x = 5, x = –2, x = 2
Question 10 of 50
2.0 Points
You have 332 feet of fencing to enclose a rectangular region. What is the maximum area?
A. 6889 square feet
B. 6885 square feet
C. 110,224 square feet
D. 27,556 square feet
Question 11 of 50
2.0 Points
Find the vertical asymptotes, if any, of the graph of the rational function.
A. x = 4 and x = 4
B. x = 4
C. x = 0 and x = 4
D. No vertical asymptote
Question 12 of 50
2.0 Points
Find the y-intercept for the graph of the quadratic function.
y + 4 = (x + 2) 2
A. (0, 4)
B. (0, 0)
C. (4, 0)
D. (0, -4)
Question 13 of 50
2.0 Points
Use Newton's Law of Cooling, T = C + (T0 – C).e kt, to solve the problem.
A cup of coffee with temperature 102°F is placed in a freezer with temperature 0°F. After 8 minutes, the temperature of the coffee is 52.5°F. What will its temperature be 13 minutes after it is placed in the freezer? Round your answer to the nearest degree.
A. 32°F
B. 29°F
C. 35°F
D. 27°F
Question 14 of 50
2.0 Points
Use the graph of f(x) = log x to obtain the graph of g(x) = log x + 5.
A.
B.
C.
D.
Question 15 of 50
2.0 Points
Evaluate or simplify the expression without using a calculator.
log 1000
A.
3
B.
30
C.
D.
Question 16 of 50
2.0 Points
A fossilized leaf contains 15% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon 14. Solve the problem.
A. 35,828
B. 15,299
C. 1311
D. 21,839
Question 17 of 50
2.0 Points
Find the exact value of the expression.
tan -1 0
A.
0
B.
C.
D.
Qu ...
5. Example 1 Use the discriminant
Equation Discriminant Number of
solutions
a. 2x2 + 6x + 5 = 0 62 – 4(2 ) (5 ) = –4 No solution
b. x2 – 7 = 0 02 – 4(1 ) ( – 7 ) = 28 Two solutions
c. 4x2 – 12x + 9 = 0 ( –12 )2 – 4(4 ) (9) = 0 One solution
6. Example 2 Multiple Choice Practice
Which statement best explains why there is only one
real solution to the quadratic equation 9x2 + 6x + 1 = 0?
The value of ( 6 )2 – 4 • 9 • 1 is positive.
The value of ( 6 )2 – 4 • 9 • 1 is equal to 0.
The value of ( 6 )2 – 4 • 9 • 1 is negative.
The value of ( 6 )2 – 4 • 9 • 1 is not a perfect
square.
7. Example 2 Multiple Choice Practice
SOLUTION
Find the value of the discriminant.
b2 – 4 • a • c = ( 6 )2 – 4 • 9 • 1 = 36 – 36 = 0
The discriminant is zero, so the equation has one real
solution.
ANSWER The correct answer is B.
8. Example 3 Find the number of x-intercepts
Find the number of x-intercepts of the graph of
y = x2 – 3x – 10.
SOLUTION
Find the number of solutions of the equation
0 = x2 – 3x – 10.
b2 – 4ac = ( – 3)2 – 4(1 ) ( –10 ) Substitute 1 for a, – 3 for b,
and –10 for c.
= 49 Simplify.
The discriminant is positive, so the equation has two
solutions. This means that the graph of y = x2 – 3x – 10
has two x-intercepts.
9. Example 3 Find the number of x-intercepts
CHECK You can use a graphing calculator to check
the answer. Notice that the graph of
y = x2 – 3x – 10 has two x-intercepts.
You can also use factoring to check the answer.
Because x2 – 3x – 10 = ( x – 5 ) ( x + 2 ), the
graph of y = x2 – 3x – 10 crosses the x-axis at
x – 5 = 0, or x = 5, and at x + 2 = 0, or x = – 2.
10. Example 4 Solve a multi-step problem
FOUNTAINS
The Centennial Fountain in Chicago shoots a water arc
that can be modeled by the graph of the equation
y = – 0.006x2 + 1.2x + 10 where x is the horizontal
distance (in feet) from the river’s north shore and y is
the height (in feet) above the river. Does the water arc
reach a height of 50 feet? If so, about how far from the
north shore is the water arc 50 feet above the water?
11. Example 4 Solve a multi-step problem
SOLUTION
STEP 1 Write a quadratic equation. You want to know
whether the water arc reaches a height of 50
feet, so let y = 50. Then write the quadratic
equation in standard form.
y = – 0.006x2 + 1.2x + 10 Write given equation.
50 = – 0.006x2 + 1.2x + 10 Substitute 50 for y.
0 = – 0.006x2 + 1.2x – 40 Subtract 50 from each side.
STEP 2 Find the value of the discriminant of
0 = – 0.006x2 + 1.2x – 40.
12. Example 4 Solve a multi-step problem
b2 – 4ac = ( 1.2)2 – 4 ( – 0.006 ) ( – 40 ) a = – 0.006, b = 1.2,
c = – 40
= 0.48 Simplify.
STEP 3 Interpret the discriminant. Because the
discriminant is positive, the equation has two
solutions. So, the water arc reaches a height of
50 feet at two points on the water arc.
STEP 4 Solve the equation 0 = – 0.006x2 + 1.2x – 40 to
find the distance from the north shore where
the water arc is 50 feet above the water.
13. Example 4 Solve a multi-step problem
–b +
– b2 – 4ac
x = Quadratic formula
2a
– 1.2 +
– 0.48
= Substitute values in
2 ( – 0.006 ) the quadratic formula.
x ≈ 42 or 158 Use a calculator.
ANSWER
The water arc is 50 feet above the water about 42 feet
from the north shore and about 158 feet from the north
shore.
14. 10.8 Warm-Up
Tell whether the equation has 2 solutions, one solution,
or no solution.
1. x 2 + 4x + 3 = 0
2. 2x 2 - 5x + 6 = 0
3. -x 2 + 2x =1
Find the number of x-intercepts of the graph of the
function.
4. y = x 2 +10x + 25
5. y = x 2 - 9x