Here are the steps to solve each equation by graphing:
1) x2 + 5x + 6 = 0
Write in standard form: x2 + 5x + 6 = 0
Graph the related function y = x2 + 5x + 6. The x-intercepts are the solutions, which are -3 and -2.
2) x2 + 8x + 16 = 0
Write in standard form: x2 + 8x + 16 = 0
Graph the related function y = x2 + 8x + 16. The x-intercepts are the solutions, which are -4 and -4.
3) x2 - 2x + 3 = 0
Write in standard
'How can we support older workers?' an ILC-UK European policy debate, support...ILC- UK
Tuesday 3rd September, M&G, Governor’s House, Laurence Pountney Hill, London, EC4R 0HH, 16:00 for a 16:30 start – 18:30
Featuring Steve Webb MP (Minister for Pensions); Christopher Brooks (Age UK) and David Sinclair (ILC-UK), presenting findings from a new policy review of European innovations in supporting longer working lives. Chaired by Baroness Greengross, CEO, ILC-UK and cross-bench peer
Europe needs older workers. Its long-term ageing population and recent economic hardships are creating huge fiscal and demographic pressures - pressures which could be greatly relieved if it can encourage its workers to remain in work for longer.
How is this to be achieved?
The European Union recently launched its Europe 2020 strategy which set employment targets of 75% for workers aged 20-64. However, with the old-age dependency ratio for the EU28 predicted to climb over 50% by 2050, much more still needs to be done.
In this event we will hear UK and EU perspectives on how older workers can be supported, with contributions from Steve Webb MP, the UK Minister for Pensions; and Christopher Brooks (Age UK)
To inform this debate, ILC-UK launched a report at the event, supported by Prudential, which shares key policy approaches being taken across to support older workers.
David Sinclair on the challenges of vaccinating adultsILC- UK
Presented at the IFA Champions Summit in early November, Director of the International Longevity Centre - UK, David Sinclair, considers the challenges and solutions to vaccinating adults.
We held an event to launch SOS 2020, supported by Aviva and Ernst and Young. This event was kindly sponsored by the Institute and Faculty of Actuaries (IFoA).
Last week the OBR Fiscal Sustainability Report noted that "public finances are likely to come under pressure over the longer term, primarily as the result of an ageing population. Under our definition of unchanged policy, the Government would end up having to spend more as a share of national income on age-related items such as pensions and health care, but the same demographic trends would leave government revenues roughly stable."
But whilst there is greater awareness of the fiscal challenges of ageing, there has been little progress in addressing an overarching plan to address the challenges. ILC-UK is launching SOS 2020 to begin to identify costed solutions to the fiscal challenges of ageing.
The House of Lords Select Committee on Public Service and Demographic Change, in its 2013 report “Ready for Ageing”, began by saying “the UK population is ageing rapidly, but we have concluded that the Government and our society are woefully underprepared.”
SOS 2020 is a major new programme of work led by ILC-UK which will raise awareness of the need to adapt our economy and society to the big strategic challenges posed by an ageing population.
SOS 2020 will outline the specific policy measures needed to achieve this goal. It will illuminate the issues that face us and develop fully considered and costed solutions that will act as a “call to action” to policy-makers and politicians. Above all SOS 2020 aims to raise national and international awareness of problems and possible solutions in which we all have a vested interest.
In an increasingly interdependent world, there is a need to look beyond national shores for arguably collective consensus and joint solutions. SOS 2020 will give us the opportunity to do this.
ILC-UK launched SOS 2020 with specific projects exploring retirement income sustainability and healthcare sustainability.
This launch event gave delegates an opportunity to feed in their thoughts on how to ensure our public policy maximises the opportunities of our ageing society.
5 Leading Strategies for Creating Credit Union Member Communications That Pro...NAFCU Services Corporation
Watch this recorded webinar to learn five strategies for taking your credit union member and prospect communications to the next level. We will show real credit union examples of these innovative enhancements that will increase opportunities for new business and members, and add value to your communications to boost ROI.
Watch the recorded webinar and learn more at http://www.nafcu.org/cathedral
'How can we support older workers?' an ILC-UK European policy debate, support...ILC- UK
Tuesday 3rd September, M&G, Governor’s House, Laurence Pountney Hill, London, EC4R 0HH, 16:00 for a 16:30 start – 18:30
Featuring Steve Webb MP (Minister for Pensions); Christopher Brooks (Age UK) and David Sinclair (ILC-UK), presenting findings from a new policy review of European innovations in supporting longer working lives. Chaired by Baroness Greengross, CEO, ILC-UK and cross-bench peer
Europe needs older workers. Its long-term ageing population and recent economic hardships are creating huge fiscal and demographic pressures - pressures which could be greatly relieved if it can encourage its workers to remain in work for longer.
How is this to be achieved?
The European Union recently launched its Europe 2020 strategy which set employment targets of 75% for workers aged 20-64. However, with the old-age dependency ratio for the EU28 predicted to climb over 50% by 2050, much more still needs to be done.
In this event we will hear UK and EU perspectives on how older workers can be supported, with contributions from Steve Webb MP, the UK Minister for Pensions; and Christopher Brooks (Age UK)
To inform this debate, ILC-UK launched a report at the event, supported by Prudential, which shares key policy approaches being taken across to support older workers.
Technology-driven change has become a constant for merchants,
financial institutions, and processors. That reality has created a shifting
landscape of new capabilities, new competitors, new rules, and new
customer expectations. It can all be complicated and confusing, but an
assessment of that landscape indicates several clear trends affecting
the industry. For more info: www.nafcu.org/vantiv
Getting More Business from Your Members with Electronic Strategies (Credit Un...NAFCU Services Corporation
In this 2012 Strategic Growth Conference session, learn how effectively communicating with your members and potential members has recently become more complicated. Explore how you can successfully integrate electronic response communications with conventional communication vehicles to stimulate your credit union’s growth. Discover the importance of your members’ data and how segmentation can make your growth strategies more valuable. You will walk away with three tools that you can start implementing immediately to make your communication strategies work better for you while developing your members’ engagement and helping your credit union grow. More info at: www.nafcu.org/cathedral
David Sinclair, Assistant Director, Policy and Communications, ILC-UK
Presentation from EngAGEd conference on Friday 5th October https://registration.livegroup.co.uk/healthyageingconference/
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
2. Vocabulary
Quadratic Equation – an equation that can be written
in the standard form ax 2 + bx + c = 0 where a ¹ 0
Zero(s) of a function – x value(s) for which y = 0
* Zero(s) of a polynomial function and root(s) of a
polynomial are the same!
3. Solve by Graphing (vs. Factoring)
Recall Solve by Factoring: x 2 - 6x + 5 = 0
(x -1)(x - 5) = 0
x =1, x = 5
Solve by Graphing:
4. Example 1 Solve a quadratic equation having two solutions
Solve x2 – 2x = 3 by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
x2 – 2x = 3 Write original equation.
x2 – 2x – 3 = 0 Subtract 3 from each side.
STEP 2 Graph the related function
y = x2 – 2x – 3 . The x-intercepts
are –1 and 3.
5. Example 1 Solve a quadratic equation having two solutions
ANSWER
The solutions of the equation x2 – 2x = 3 are – 1 and 3.
CHECK You can check –1 and 3 in the original equation.
x2 – 2x = 3 x2 – 2x = 3 Write original equation.
?
( – 1)2 – 2 (– 1) = 3 ?
( 3)2 – 2( 3) = 3 Substitute for x.
3 = 3 3 = 3 Simplify. Each solution
checks.
6. Example 2 Solve a quadratic equation having one solution
Solve – x2 + 2x = 1 by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
– x2 + 2x = 1 Write original equation.
– x2 + 2x – 1 = 0 Subtract 1 from each side.
STEP 2 Graph the related function y = – x2 + 2x – 1 .
The x-intercept is 1.
7. Example 2 Solve a quadratic equation having one solution
ANSWER
The solution of the equation – x2 + 2x = 1 is 1.
8. Example 3 Solve a quadratic equation having no solution
Solve x2 + 7 = 4x by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
x2 + 7 = 4x Write original equation.
x2 – 4x + 7 = 0 Subtract 4x from each side.
STEP 2 Graph the related function
y = x2 – 4x + 7. The graph has
no x-intercepts.
9. Example 3 Solve a quadratic equation having no solution
ANSWER
The equation x2 + 7 = 4x has no solution.
10. Number of Solutions of a Quadratic Equation
Two Solutions One Solution No Solution
A quadratic equation A quadratic equation A quadratic equation
has two solutions if has one solution if the has no real solution if
the graph of its related graph of its related the graph of its related
function has two x- function has one x- function has no x-
intercepts. intercept. intercepts.
11. Example 4 Multiple Choice Practice
The graph of the equation
y = x2 + 6x – 7 is shown. For
what value or values of x is y = 0?
x = –7 only x = 1 only
x = – 7 and x = 1 x = –1 and x = 7
SOLUTION
You can see from the graph that the x-intercepts are –7
and 1. So, y = 0 when x = –7 and x = 1.
ANSWER The correct answer is C.
12. Example 5 Approximate the zeros of a quadratic function
Approximate the zeros of y = x2 + 4x + 1 to the nearest
tenth.
SOLUTION
STEP 1 Graph the function y = x2 + 4x + 1. There are
two x-intercepts: one between – 4 and –3 and
another between –1 and 0.
13. Example 5 Approximate the zeros of a quadratic function
STEP 2 Make a table of values for x-values between
– 4 and – 3 and between – 1 and 0 using an
increment of 0.1. Look for a change in the
signs of the function values.
x – 3.9 – 3.8 – 3.7 – 3.6 – 3.5 – 3.4 – 3.3 – 3.2 – 3.1
y 0.61 0.24 – 0.11 – 0.44 – 0.75 – 1.04 – 1.31 – 1.56 – 1.79
x – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1
y – 1.79 – 1.56 – 1.31 – 1.04 – 0.75 – 0.44 – 0.11 0.24 0.61
14. Example 5 Approximate the zeros of a quadratic function
ANSWER
In each table, the function value closest to 0 is – 0.11.
So, the zeros of y = x2 + 4x + 1 are about – 3.7 and
about – 0.3.
15. Example 6 Solve a multi-step problem
SPORTS
An athlete throws a shot put with
an initial vertical velocity of 40 feet
per second.
a. Write an equation that models
the height h (in feet) of the
shot put as a function of the
time t (in seconds) after it is
thrown.
b. Use the equation to find the time that the shot put is
in the air.
16. Example 6 Solve a multi-step problem
SOLUTION
a. Use the initial vertical velocity and height to write a
vertical motion model.
h = – 16t2 + vt + s Vertical motion model
Substitute 40 for v and
h = – 16t2 + 40t + 6.5 6.5 for s.
b. The shot put lands when h = 0. To find the time t
when h = 0, solve 0 = –16t 2 + 40t + 6.5 for t.
To solve the equation, graph the
related function h = –16t2 + 40t + 6.5
on a graphing calculator. Use the
trace feature to find the t-intercepts.
17. Example 6 Solve a multi-step problem
ANSWER
There is only one positive t-intercept. The shot put is in
the air for about 2.6 seconds.
18. Relating Roots of Polynomials,
Solutions of Equations, x-intercepts
of Graphs, and Zeros of Functions
The Roots of the Polynomial -x 2 +8x -12 are 2 & 6.
The Solutions of the equation -x 2 +8x -12 = 0 are
2 & 6.
The x-intercepts of the graph of y = -x +8x -12 occur
2
where y=0, so the x-intercepts are 2 and 6.
The Zeros of the function y = -x +8x -12 are the
2
values of x for which y=0, so the zeros are 2 and 6.
19. 10.4 Warm-Up
Solve the equation by graphing.
1. x 2 + 5x + 6 = 0
2. x 2 +8x +16 = 0
3. x 2 - 2x + 3 = 0