Quadratic Equations Quadratic Equations

7/29/2019 Quadratic Equations Quadratic Equations

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Solving

Equations


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A quadratic equation is an equation equivalent to one of the form

Where a, b, and c are real numbers and a 0

02 cbxax

To solve a quadratic equation we get it in the form above

and see if it will factor.652 xx Get form above by subtracting 5x andadding 6 to both sides to get 0 on right side.

-5x + 6 -5x + 6

0652

xx Factor.

023 xx Use the Null Factor law and set eachfactor = 0 and solve.

02or03 xx 3x 2x

So if we have an equation inxand the highest power is 2, it is quadratic.


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In this form we could have the case where b = 0.

02 cbxax

Remember standard form for a quadratic equation is:

02 cax

002 cxax

When this is the case, we get the x2 alone and then square

root both sides.

062 2 x Getx2 alone by adding 6 to both sides and then

dividing both sides by 2+ 6 + 6

62

2

x2 2 32 x

Now take the square root of both

sides remembering that you must

consider both the positive andnegative root.

3xLet's

check: 06322

06322

066 066

Now take the square root of both

sides remembering that you must

consider both the positive andnegative root.


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02 cbxax

What if in standard form, c = 0?

002 bxax

We could factor by pulling

anxout of each term.

032 2 xx Factor out the commonx

032 xx Use the Null Factor law and set eachfactor = 0 and solve.

032or0 xx

2

3or0 xx

If you put either of these values in forx

in the original equation you can see it

makes a true statement.


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02 cbxaxWhat are we going to do if we have non-zero values for

a, b and c but can't factor the left hand side?

0362 xxThis will not factor so we will complete the

square and apply the square root method.

First get the constant term on the other side by

subtracting 3 from both sides.36

2 xx

___3___62 xx

We are now going to add a number to the left side so it will factor

into a perfect square. This means that it will factor into two

identical factors. If we add a number to one side of the equation,

we need to add it to the other to keep the equation true.

Let's add 9. Right now we'll see that it works and then we'll look at howto find it.

9 9 6962 xx


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6962 xx Now factor the left hand side.

633 xx

two identical factors

63 2 xThis can be written as:

Now we'll get rid of the square by

square rooting both sides.

63 2 xRemember you need both the

positive and negative root!

63x Subtract 3 from both sides to getxalone.

63 xThese are the answers in exact form. We

can put them in a calculator to get two

approximate answers.

55.063 x 45.563 x


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Okay---so this works to solve the equation but how did we

know to add 9 to both sides?

___3___62

xx 9 9

633 xx We wanted the left hand side to factorinto two identical factors.

When you FOIL, the outer terms and theinner terms need to be identical and need

to add up to 6x.+3 x+3

x

6 x

The last term in the original trinomial will then be the middleterm's coefficient divided by 2 and squared since last term

times last term will be (3)(3) or 32.

So to complete the square, the number to add to both sides

is

the middle term's coefficient divided by 2 and squared


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By completing the square on a general quadratic equation in

standard form we come up with what is called the quadratic formula.

(Remember the song!! )

a

acbbx

2

42

This formula can be used to solve any quadratic equationwhether it factors or not. If it factors, it is generally easier to

factor---but this formula would give you the solutions as well.

We solved this by completing the square

but let's solve it using the quadratic formula

a

acbbx

2

42

1

(1)

(1)

6 6 (3)

2

12366

Don't make a mistake with order of operations!Let's do the power and the multiplying first.

02 cbxax

036

2 xx


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2

12366 x

2

246

626424

2

626

2

632

There's a 2 in common in

the terms of the numerator

63 These are the solutions wegot when we completed the

square on this problem.

NOTE: When using this formula if you've simplified under the

radical and end up with a negative, there are no real solutions.

(There are complex (imaginary) solutions, but that will be dealt

with in year 12 Calculus).


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SUMMARY OF SOLVING QUADRATIC EQUATIONS

Get the equation in standard form: 02 cbxax

If there is no middle term (b = 0) then get thex2 alone and square

root both sides (if you get a negative under the square root there are

no real solutions).

If there is no constant term (c = 0) then factor out the commonxand use the null factor law to solve (set each factor = 0).

Ifa, band care non-zero, see if you can factor and use the null

factor law to solve.

If it doesn't factor or is hard to factor, use the quadratic formula

to solve (if you get a negative under the square root there are no real

solutions).


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a

acbbxcbxax

2

40

22

If we have a quadratic equation and are considering solutions

from the real number system, using the quadratic formula, one of

three things can happen.

3. The "stuff" under the square root can be negative and we'd get

no real solutions.

The "stuff" under the square root is called the discriminant.

This "discriminates" or tells us what type of solutions we'll have.

1. The "stuff" under the square root can be positive and we'd gettwo unequal real solutions 04if 2 acb

2. The "stuff" under the square root can be zero and we'd get one

solution (called a repeated or double root because it would factor

into two equal factors, each giving us the same solution).04if

2

acb

04if2

acb

The Discriminant acb 42


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5-2 Two-Way Selection

The decision is described to the computer as aconditional statement that can be answered either true

or false. If the answer is true, one or more action

statements are executed. If the answer is false, then a

different action or set of actions is executed.

ifelse andNull else Statement

Nested ifStatements and Dangling else Problem

Simplifying ifStatements

Conditional Expressions

Topics discussed in this section:


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Computer Science: AStructured

14

FIGURE 5-6 Two-way Decision Logic


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FIGURE 5-7 if...else Logic Flow


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Table 5-2 Syntactical Rules forifelse Statements


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FIGURE 5-8 A Simple if...else Statement


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FIGURE 5-9 Compound Statements in an if...else


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5-3 Multiway Selection

In addition to two-way selection, most programming

languages provide another selection concept known as

multiway selection. Multiway selection chooses among

several alternatives. C has two different ways toimplement multiway selection: the switch statement and

else-if construct.

Theswitch Statement

The else-if

Topics discussed in this section:


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FIGURE 5-19switch Decision Logic


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FIGURE 5-24 The else-ifLogic Design for Program 5-9


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The else-ifis an artificial C construct that is only used when

1. The selection variable is not an integral, and

2. The same variable is being tested in the expressions.

Note


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General Syntax of else-if

constructif (expression-1)

statement-1;

else if (expression-2)statement-2;

..

..else

statement-n;

#i l d < tdi h>


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#include

#include

#include

#include

void main()

{float a,b,c,x1,x2,disc;

clrscr();

printf("Enter the co-efficients\n");

scanf("%f%f%f",&a,&b,&c);

if(a==0)

{printf( equation is not quadratic \n);

exit(0);

}

disc=b*b-4*a*c;/*to find discriminant*/

if(disc>0) /*distinct roots*/

{x1=(-b+sqrt(disc))/(2*a);

x2=(-b-sqrt(disc))/(2*a);

printf("The roots are distinct\n");

}


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else

{

x1=-b/(2*a);/*complex roots*/

x2=sqrt(fabs(disc))/(2*a);

printf("The roots are complex\n");

printf("The first root=%f+i%f\n",x1,x2);printf("The second root=%f-i%f\n",x1,x2);

getch();

}

} /* end of main function */

else if(disc==0)/*Equal roots*/{

x1=x2=-b/(2*a);

printf("The roots are equal\n");

printf("x1=%f\nx2=%f\n",x1,x2);

}


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Quadratic Equations Quadratic Equations