It is not always possible to solve a quadratic equation by factorising and sometimes it is lengthy and tedious to solve a quadratic equation by completing the square. In these situations, you can use the
quadratic formula that gives the solutions to any quadratic equation.
Consider the general form of the quadratic function:
Factor out the
to get:
Now we need to do some detective work to figure out how to turn
[link] into a perfect square plus some extra terms. We know that for a perfect square:
and
The key is the middle term on the right hand side, which is
the first term
the second term of the left hand side.
In
[link] , we know that the first term is
so 2
the second term is
. This means that the second term is
. So,
In general if you add a quantity and subtract the same quantity, nothing has changed. This means if we add and subtract
from the right hand side of
[link] we will get:
We set
to find its roots, which yields:
Now dividing by
and taking the square root of both sides gives the
expression
Finally, solving for
implies that
which can be further simplified to:
These are the solutions to the quadratic equation. Notice that there are two
solutions in general, but these may not always exists (depending on the sign ofthe expression
under the square root). These solutions are also
called the
roots of the quadratic equation.
Find the roots of the function
.
The expression cannot be factorised. Therefore, the general quadratic formula must be used.
From the equation:
Always write down the formula first and then substitute the values of
and
.
The two roots of
are
and
.
Find the solutions to the quadratic equation
.
The expression cannot be factorised. Therefore, the general quadratic formula must be used.
From the equation:
Since the expression under the square root is negative these are not
real solutions (
is not a real number). Therefore
there are no real solutions to the quadratic equation
. This means that the graph of the quadratic function
has no
-intercepts, but that the entire graph lies above the
-axis.
Solution by the quadratic formula
Solve for
using the quadratic formula.
In all the examples done so far, the solutions were left in surd form. Answers can also be given in decimal form, using the calculator. Read the instructions when answering questions in a test or exam whether to leave answers in surd form, or in decimal form to an appropriate number of decimal places.
Completing the square as a method to solve a quadratic equation is only done when specifically asked.
Mixed exercises
Solve the quadratic equations by either factorisation, completing the square or by using the quadratic formula:
Always try to factorise first, then use the formula if the trinomial cannot be factorised.
Do some of them by completing the square and then compare answers to those done using the other methods.
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Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product