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Equations And Inequalities

 In this article, we are going to learn about Equalities and Inequalities:

  • Relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations
  • Related conditions for a given line to
    • intersect a given curve
    • be a tangent to a given curve
    • not intersect a given curve

This article is specifically written to meet the requirements of Secondary 3 Additional Mathematics.

Quadratic Formula

The general solution of the equation ax2+bx+c=0  where a0, are solved using the quadratic formula, x=b±b24ac2a  where x is called the roots of the equation. We will be using this to understand the nature of the roots.

Discriminant and Nature of Roots

A discriminant, b24ac, is a part of the quadratic formula which can be used to tell us the number of roots a quadratic equation has and nature refers to the type of roots that exist.

For the quadratic formula x=b±b24ac2a

b24ac

Roots 

Nature Of Roots

>0

x=b+b24ac2a

x=bb24ac2a

2 real & distinct roots

=0

x=b±02a=b2a

2 real & repeated root 

<0

x=b±-ve2a

Roots are

  • Imaginary 
  • Complex
  • No real roots

Solving Linear Simultaneous Equations

Question 1:

Solve these simultaneous equations.

y=2x1y=x+5

Solution:

You can solve this using the Substitution Method.

y=2x1(1)y=x+5(2) 

Since both equations equate to y hence you can substitute the value of (1) in (2)    

2x1=x+52x+x=1+53x=6x=2 is the root

 

 

Solving Linear & Non–Linear Simultaneous Equation

Question 2:

Solve these simultaneous equations

y=x2+2x5y=x+5

Solution: 

You can solve this using the Substitution Method.

y=x2+2x5(1)y=x+5(2)

Since both equations equate to y hence you can substitute the value of (1) in (2)

x+5=x2+2x5x2+2x5+x5=0x2+3x10=0

 

x=(3)±324(1)(10)2(1)x=3±492x=2or5

Thus, the discriminant b24ac=49 which is >0 and hence there are two real roots, 2 and 5.

 

Question 3:

Solve these simultaneous equations

y=x211x+30y=x+5

Solution:

You can solve this using the Substitution Method.

y=x211x+30(1)y=x+5(2)

Since both equations equate to y hence you can substitute the value of (1) in (2)

x211x+30=x+5x211x+30+x5=0x210x+25=0

 

x=(10)±(10)24(1)(25)2(1)x=10±02x=5

Thus, the discriminant is b24ac=0 which is =0 and hence there is one real distinct root, 5.

 

Question 4:

Solve these simultaneous equations

y=x2x+6y=x+5

Solution:

You can solve this using the Substitution Method.

y=x2x+6(1)y=x+5(2)

Since both equations equate to y hence you can substitute the value of (1) in (2).

x2x+6=x+5x2x+6+x5=0x2+0x+1=0

x=(0)±(0)24(1)(1)2(1)x=(0)+42x=(0)42

As the discriminant b24ac=4  which is <0 and hence there are no real distinct roots or no solution possible.

Related conditions for determining the number of Points of Intersection of a Line and a Curve

When two equations are given in the form

y=px2+qx+ry=ax+b

You can solve them using x=b±b24ac2a

Example 1:

Find the value of m for which the line y=mx3 is a tangent to the curve y=x+1x.

Solution:

When we look at the question, it mentions a tangent to the curve y=x+1x which means the curve gets cut only at one point. It means that b24ac=0 and there is only one real root.

y=mx3(1)y=x+1x(2)

 

mx3=x+1xmx3=x2+1xmx23x=x2+1x2+1mx2+3x=0(1m)x2+3x+1=0

Now we have  a=(1m),b=3,c=1

We already know b24ac=0

Hence, 

324(1m)(1)=0(1m)=94m=194m=54

 

Question 5:

Find the value of m for which the line y=x+m is a tangent to the curve y=2x21.

Solution: 

When we look at the question, it mentions a tangent to the curve y=2x21, which means the curve gets cut only at one point. It means that b24ac=0 and there is only one real root.

Equating both equations with each other we get,

x+m=2x212x2xm1=02x2x(m+1)=0

Hence, we obtain a=2,b=1andc=(m+1)

(1)24(2)(m1)=018(m1)=01+8m+8=08m+9=0m=98m=118

Conclusion

In this article, we have observed the relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations as per the Secondary 3 Additional Mathematics syllabus in Singapore.

We have also studied the related conditions for a given line to

  • intersect a given curve
  • be a tangent to a given curve
  • not intersect a given curve

Multiple examples and questions are also given to aid in understanding these concepts better. Keep learning! Keep improving! 

 

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