Find Where Undefined/Discontinuous (y+7)/(4y^2+3y-1)

$\frac{y + 7}{4 y^{2} + 3 y - 1}$

Step 1

Set the denominator in $\frac{y + 7}{4 y^{2} + 3 y - 1}$ equal to $0$ to find where the expression is undefined.

$4 y^{2} + 3 y - 1 = 0$

Step 2

Solve for $y$ .

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = 3$ .

Factor $3$ out of $3 y$ .

$4 y^{2} + 3 (y) - 1 = 0$

Rewrite $3$ as $- 1$ plus $4$

$4 y^{2} + (- 1 + 4) y - 1 = 0$

Apply the distributive property.

$4 y^{2} - 1 y + 4 y - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 y^{2} - 1 y) + 4 y - 1 = 0$

Factor out the greatest common factor (GCF) from each group.

$y (4 y - 1) + 1 (4 y - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 y - 1$ .

$(4 y - 1) (y + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 y - 1 = 0$

$y + 1 = 0$

Set $4 y - 1$ equal to $0$ and solve for $y$ .

Set $4 y - 1$ equal to $0$ .

$4 y - 1 = 0$

Solve $4 y - 1 = 0$ for $y$ .

Add $1$ to both sides of the equation.

$4 y = 1$

Divide each term in $4 y = 1$ by $4$ and simplify.

Divide each term in $4 y = 1$ by $4$ .

$\frac{4 y}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{1}{4}$

Divide $y$ by $1$ .

$y = \frac{1}{4}$

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

The final solution is all the values that make $(4 y - 1) (y + 1) = 0$ true.

$y = \frac{1}{4}, - 1$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$y = - 1, y = \frac{1}{4}$

Step 4

Find Where Undefined/Discontinuous (y+7)/(4y^2+3y-1)

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