A fraction is said to be undefined if the denominator is zero, e.g 9/0, 2/0, 3/0 etc.

For instance, 1/x+2 is undefined

If the value of x = -2

When x = -2,

Then 1/x+2 = 1/-2+2 = 1/0

Division by zero is in possible therefore, the fraction 1/x+2 is said to be undefined when x=-2

Given a fraction Nx/Dx where Nx is the numerator and Dx the denominator, the fraction Nx/Dx will be said to be undefined if the denominator Dx is equal to zero.

Now that we are conversant with the procedure to be used when dealing with undefined fraction, let solve some important problems.

**Example 1:**

Find the values of x for which the fraction x+2/x^{2}-3x-10 is undefined

Solution

Recall that a fraction is said to be undefined if the denominator is equal to zero

That means

*x ^{2}-3x=10 – 0*

Since it is a quadratic equation, we need to solve by factorization

*x ^{2}-3x=10 – 0*

The last and first terms are *-10* and *x ^{2}*

*-10 *x *x ^{2} = -10x^{2}*

The middle term is -3x

Two factorization in x whose sum will give -3x and product will give -10x^{2} are -5x and 2x

We replace -3x with -5x and 2x

*x2 + 2x-5x-10 = 0*

*x(x+2)-5(x+2) = 0*

*(x+2) (x-5) = 0*

*x + 2 = 0 *and *x – 5 = 0*

*x =2 *or *x =5*

The fraction is undefined which *x = -2 or x = 5*

**Example 2**

What are the values of x for which the expression

*6x + 1/x ^{2}-18x-20* is not defined?

Solution

X will be undefined when *x2+8x-20 = 0*

We solve again by factorization

The first term is x^{2} and the last term is -20

The product is *x ^{2}– 20 = -20x^{2}*

The middle term = +8x

The two factors in 8 whose sum gives +8x and product gives -20x^{2} are *10x and -2x*

*(x ^{2}+10x)-(2x-20) = 0*

*x(x +10)-2 (x+10) = 0*

*(x+10) (x+2) = 0*

*x + 10 = 0 and x -2 =0*

*x = -10 or x =2*

Therefore, the expression is undefined if *x = 10 or x =2*

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**Example 3:**

- For what values of x is the expression

*2x-16/x ^{2}-11x+24*

- For what value of x is the expression equal to zero

Solution

- The expression is undefined when its denominator

*x ^{2}=11x+24 is zero*

*i.ex ^{2}-11x+24 = 0*

The first term is x^{2}and the last term is 24

The product yield *x ^{2}x24 = 24x^{2}*

The middle term is -11x

Two factors in x whose sum gives -11x and product gives 24x^{2} are -8x and -3x

We the replace -11x with the factors

*x ^{2}-8x-3x+24=0*

*x(x-8)-3(x-8) =0*

*(x-8) (x-3) = 0*

*x-8 = 0 or x – 3 =0*

therefore, the expression is undefined of x = 8 or x = 0

- The expression is zero which the algebraic expression is equal to zero i.e

*2x+16=0*

*2x/2 = 16/8*

*x = 8*

the expression will be zero when x = 8

**Example 4:**

Find the value of x for which the algebraic fraction is undefined

*9/x + x ^{2}-1/x^{2}-16*

**Solution**

Multiply both sides of the expression by the L.C.M *x(x ^{2}-16)*

*x(x ^{2}-16) *x

*9/x + x(x*

^{2}-16) x^{2}-1/x^{2}-10*9(x ^{2}-16) +x(x^{2}-1)*

But the expression will be undefined when

*x ^{2}=16 = 0*

*x ^{2}=16*

Take square root of both sides

*x ^{2}*

^{x1/2 }

*=*

*x = ± 4*

which means x = -4 or x = 4 therefore the expression is undefined when x = 4 or -4 or 0

**Example 5:**

- For what values of x is the expression

*2(x-1) /x ^{2}– 9 *undefined

- For what values of A is the expression zero?

Solution

- The expression is undefined when its denominator

*x ^{2} – 9 is zero*

*i.ex ^{2} – 9 = 0*

*x =9*

take the square roots of both sides

*x ^{2}*

^{x½ }

*=*

*x = ± 3*

That means x = 3 or -3, therefore the fraction is undefined when x = -3 or 2

- The expression is zero when the algebraic expression, (the numerator) is equal to zero

i.e*3(x-1) = 0*

*3x – 3 = 0*

*3x/3 = 3/3*

*x = 1*

the expression will be zero when x = 1

If you have any questions about undefined fraction, kindly let us know in the comment box.