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Different methods for solving quadratic equations

6.5  Equations of the Form ax^2 + bx + c = 0

  By a quadratic equation in the single variable x, we mean any equation that can be transformed through elementary transformations to an equation of the form

  ax^2 + bx + c = 0,  a!=0

  When the equation is written in the above form we will say that it is in standard form.

  The first type of quadratic equation that we will consider is an equation of the form


  We first solve for x^2 to obtain


  Next we solve for x by taking the square root to obtain


  The solution set is therefore {root((-c/a)),-root((-c/a))}.

Example 1.  Solve 9x^2-25=0





  The solution set is {5/3,-5/3}.

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6.6  Solution of ax^2 + bx + c = 0 by Factoring

  We recall from Chapter 1 the important fact that if A and B are real numbers and



  A=0  or  B=0

  This property is the key to solving quadratic equations by the method of factoring. It is important to note that the above reasoning applies only when the right-hand side is zero; no other real number will do.

Example 1.  Solve the equation x^2-4x-5=0.




  x-5=0  or x+1=0

  so the solution set is {5,-1}

Example 2  Solve the equation 2x^2-5x=3.

  We add -3 to both sides to obtain 0 on the right-hand side.





  2x+1=0  or  x-3=0

  so the solution set is


  More generally if



  A=0, B=0,...  or C=0

  Consequently this method of solution will apply to any degree equation as long as we are able to factor the equation into linear terms. For example, if we have



  x=0x-3=0, or x+1=0

  and hence the solution set is 


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6.7  Completing the Square

  As we have seen, quadratic equations of the form


  where amay be solved by the method of extraction of roots. In fact, since every real number a has two square roots (either real or imaginary) we have two solutions. These are given by is any real number, 

  x=root(a)  or  x=-root(a)

  For example , from the equation


  we have

  x=root(25)=5  or  x=-root(25)=-5

  From the equation


  we have

  x=root(-9)=3root(-1)=3i  or  x=-root(-9)=-3i

  Equations of the form


  can be solved by the same method. For example


  implies that

  x-3=5  or  x-3=-5

  from which we obtain

  x=8  or  x=-2

  In order to write x^2+-2ax+b=0 in the form


  we note that


  Hence, in order to obtain a perfect square on the left-hand side when the first two terms x^2+-2ax are given, we must add a^2, which is the square of half the coefficient of the x term, to both sides of the equation. This method is called the method of completing the square. Note that to complete the square the coefficient of x^2 must be 1.

Example 1.  Solve the equation x^2-10x-24=0by completing the square. We first add 24 to both sides to obtain 




  We now extract the square roots to obtain

  x-5=root(47)=7  or  x-5=-root(49)=-7


  x=12  or  x=-2

  Therefore, the solution set is {12, -2}.

Example 2.   Solve 4x^2+4x-3=0.

  we add 3 to both sides and then divide by 4 to obtain


  Adding (1/2)^2=1/4to both sides we obtain




  Extracting roots we obtain

  x+1/2=1  or  x+1/2=-1.

  Thus the solution set is {1/2, 3/2}.

  The method of completing the square has other applications.

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Example 3.  Write x^2+y^2-4x+6y-3=0 in the form (x+-h)^2+(y+-k)^2=r^2

  We group the x terms and the y terms obtaining quadratic expressions in x and y. We then complete the square in each expression.





Example 4.  Write y=x^2+2x+3 in the form y+-k=(x+-h)^2.





6.8  The Quadratic Formula

  We can apply the methods of completing the square to the general quadratic equation

  ax^2+bx+x=0,  a!=0

  Add -c to both sides.


  Divide sides by a.


  Add (b/(2a))^2=(b^2)/(4a^2) to both sides.



  Take the square root.






  This formula, called the quadratic formula, gives both solutions of the general quadratic equation expressed in terms of the coefficients a, b, and c. The symbol ± is used to condense the two equations.


  The solution set of the general quadratic equation is


  In considering the quadratic formula we can see that the nature of the solutions-complex, irrational, or rational-depends entirely on the term under the radical, namely,


  This term is called the discriminant of the quadratic equation.

  1. The solutions are real if b^2-4ac>=0 and complex if b^2-4ac<0.

  2. If a, b, and c are rational and b^2-4ac>=0, then the solutions are rational if and only if b^2-4ac is a perfect square.

Example 1.  Solve the equation 3x^2 + x - 2 = 0 by the quadratic formula.

  Substitute a=3b=1,c=-2in the quadratic formula.






Hence the solution set is {2/3,-1}

Example 2.  Solve the equation x^2+4x+16=0.

  We have a=1b=4, and c=16.





  Thus, the solution set is {-2+2root(3i),− 2-2 √3 i}.

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  In Chapter 3 we were able to factor some polynomials of second degree with integral coefficients. We are now in a position to consider the problem of factoring any second degree polynomial where the coefficients may now be real numbers. We must allow the use of complex numbers in the factors.

  Consider the general second degree polynomial and use the method of completing the square.








  If we donate (-b+root(b^2-4ac))/(2a) by r_1 and (-b-root(b^2-4ac))/(2a) by r_2, then we have


  We observe that the numbers r_1, and r_2 are just the solutions of (ax^2+bx+c=0). These solutions, r_1 and r_2, are called the roots of the polynomial ax^2+bx+c. Thus, to factor ax^2+bx+c, we obtain its roots by solving the polynomial equation (ax^2+bx+c=0).

Example 3.  Factor 3x^2+2x+1.

  We find the two roots of the polynomial, that is, the solutions of the polynomial equation





  Since the two solutions are (-1+root(2i))/3 and (-1-root(2i))/3, we have


  If an equation in more than one variable is a quadratic equation in one of its variables, then the equation can be solved for that variable by using the quadratic formula.

Example 4.  Solve x^2+5+2y=-7xy for x.



  As a quadratic in x we have a=1,b=7y, and c=2y+5. Therefore,



6.9  Problems Leading to Quadratics

  By various techniques certain types of equations lead to quadratic equations. Often these techniques produce equations that are not equivalent to the original ones. The steps in the solution of a given problem should be carefully reviewed to see if the replacement set changes at a given step. If so, we must make sure that the solutions of the final equation are solutions of the original equation.

Example 1.  Solve the equation (x+5)/(x+3)-(x+6)/x=2/3.

  Note that x!=0,3. Multiply by the L.C.D. 3x(x-3).


  Solve the equation 






  The solution set of the last equation is {9,-3}. This is also the solution set of the original equation, since the only numbers not in the replacement set of the original equation are 0 and 3, neither of which is an element of the solution set of the last equation.

Example 2.  Find all real solutions of the equation root(2x+5)=x+1.

  Square both sides.


  The replacement set of this equation is the set of all real numbers, while the replacement set of the original equation is not the set of all real numbers. Since the operation of squaring enlarges the replacement set, the solution set may also be enlarged. Therefore we must check the solutions that we obtain for the second equation in the original equation.





  The possible solution are 2 and -2.




  Therefore, 2 is a solution.




  Therefore, -2 is not a solution. The solution set is {2}.

Example 3.  Solve the equation (x^2-4)^2-5(x^2-4)+6=0.


  Let A=x^2-4




  A=3,  A=2

  Since A=x^2-4

  x^2-4=3  or  x^2-4=2

  both of which we now solve.







  The solution set is therefore {root(7),-root(7),root(6),-root(6)}.

Example 4.  Find all real solution of Z^(3/2)-7Z^(3/4)-8=0.


  Let B=Z^(3/4),  then B^2=Z^(3/2).




  B=8  or  B=-1

  Since B=Z^(3/4),

  Z^(3/4)=8  or  Z^(3/4)=-1

  Solving, we have








  The solution Z = 1, however, does not satisfy the equation Z^(3/4)=-1. In fact we could have observed that this equation has no real solution, since for negative real numbers Z,Z^(3/4) is not defined, while for non-negative Z,Z^(3/4) is non-negative. It is easy to see that Z= 16 does satisfy our original equation, and hence the solution set is {16}.

Example 5.  Solve x^2+125=0.





  x+5=0  or  x^2-5x+25=0








  Hence the solution set is {-5,5/2+(5root(3)/2)i,5/2-(5root(3)/2)i}.

Example 6.  Solve root(2x+4)=root(x)-2.


  Square both sides.




  Square both sides.





  x=0  or  x=16




  Therefore, 0 is not a solution.





  Therefore, 16 is not a solution. Since 0 and 16 are the only possible solutions, the original equation has no solution. In case an equation has no solutions we use the symbol {empty} to represent the solution set, that is, the set with no elements. The set {empty} is called the empty set.

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6.10  Statement Problems of the Quadratic Type

  Our method of approach will be the same as in Section 6.4, only here our equation will be one that yields a quadratic equation in a single variable.

Example 1.  Find two numbers whose sum is 8 and whose product is 12.
Step 1. Reread!

Step 2. Let x be one of the numbers.

Step 3. Since their product is 12 the other number is 12/x.

Step 4. The sum of the two numbers is 8.

Step 5. x+12/x=8

Step 6. Solve. 





  x=6  or  x=2

  In either case the two numbers are 6 and 2.

Example 2. If the perimeter of a rectangle is 44 ft while the area is 120 ft^2,what are its dimensions?

Step 1. Reread!

Step 2. Let x be the length of one side.

Step 3. Since 1/2 of the perimeter is 22 ft the other side is 22 - x.

      statement problem of quadratic type

Step 4. The area of the rectangle is 120 ft^2.

Step 5. x(22-x)=120

Step 6. Solve.




  x=10  or  x=12

  In either case the two sides are 10 ft  and 12 ft in length.

Example 3. Within a rectangular garden 18 yards long and 10 yards wide we are to pave a border of uniform width. If we are to leave an area of 84 square yards for flowers, how wide should the walk be?

Step 1. Reread!

Step 2. Let x be the width of the walk.

Step 3. The dimensions of the inside area would be 18-2x and 10 -2x.

      statement problem of quadratic type - 1   

Step 4. The inside area is to be 84 sq. yds.

Step 5. (18-2x)(10-2x)=84

Step 6. Solve.





  x=12  or  x=2

  Clearly the width of the walk would be 2 yards and not 12 yards.

Example 4. If Joe takes 6 hours longer than his father to assemble a machine and together they can assemble it in 4 hours, how long would it take each of them working alone to assemble the machine?

Step 1. Reread!

Step 2. Let x be the number of hours it would take the father to assemble the machine.

Step 3. It would take x+6 hours for Joe to assemble the machine. Choosing one hour as our unit of time, 1/x would be the amount the father could do in one hour, while 1/4 would be the amount that they could do together in one hour.
Step 4. The amount that Joe can do in one hour plus the amount that his father can do in one hour would be equal to 1/4.

Step 5. 1/(x+6)+1/x=1/4.

Step 6. Solve. Multiply by the L.C.D. 4x(x+6).





  x=6  or  x=-4

  Clearly x = -4 is not a possibility and therefore it would take Joe 12 hours and his father 6 hours.