Algebra Examples

y^{3} - 27 = 9 y^{2} - 27 y

$y^{3} - 27 = 9 y^{2} - 27 y$

Step 1

Solve for

y

$y$ .

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Step 1.1

Since

y

$y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

9 y^{2} - 27 y = y^{3} - 27

$9 y^{2} - 27 y = y^{3} - 27$

Step 1.2

Subtract

y^{3}

$y^{3}$ from both sides of the equation.

9 y^{2} - 27 y - y^{3} = - 27

$9 y^{2} - 27 y - y^{3} = - 27$

Step 1.3

Add

27

$27$ to both sides of the equation.

9 y^{2} - 27 y - y^{3} + 27 = 0

$9 y^{2} - 27 y - y^{3} + 27 = 0$

Step 1.4

Factor the left side of the equation.

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Step 1.4.1

Reorder terms.

- y^{3} + 9 y^{2} - 27 y + 27 = 0

$- y^{3} + 9 y^{2} - 27 y + 27 = 0$

Step 1.4.2

Factor

- y^{3} + 9 y^{2} - 27 y + 27

$- y^{3} + 9 y^{2} - 27 y + 27$ using the rational roots test.

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Step 1.4.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form

\frac{p}{q}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 27, \pm 3, \pm 9

$p = \pm 1, \pm 27, \pm 3, \pm 9$

q = \pm 1

$q = \pm 1$

Step 1.4.2.2

Find every combination of

\pm \frac{p}{q}

$\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

\pm 1, \pm 27, \pm 3, \pm 9

$\pm 1, \pm 27, \pm 3, \pm 9$

Step 1.4.2.3

Substitute

3

$3$ and simplify the expression. In this case, the expression is equal to

0

$0$ so

3

$3$ is a root of the polynomial.

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Step 1.4.2.3.1

Substitute

3

$3$ into the polynomial.

- 3^{3} + 9 \cdot 3^{2} - 27 \cdot 3 + 27

$- 3^{3} + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.2

Raise

3

$3$ to the power of

3

$3$ .

- 1 \cdot 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27

$- 1 \cdot 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.3

Multiply

- 1

$- 1$ by

27

$27$ .

- 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27

$- 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.4

Raise

3

$3$ to the power of

2

$2$ .

- 27 + 9 \cdot 9 - 27 \cdot 3 + 27

$- 27 + 9 \cdot 9 - 27 \cdot 3 + 27$

Step 1.4.2.3.5

Multiply

9

$9$ by

9

$9$ .

- 27 + 81 - 27 \cdot 3 + 27

$- 27 + 81 - 27 \cdot 3 + 27$

Step 1.4.2.3.6

Add

- 27

$- 27$ and

81

$81$ .

54 - 27 \cdot 3 + 27

$54 - 27 \cdot 3 + 27$

Step 1.4.2.3.7

Multiply

- 27

$- 27$ by

3

$3$ .

54 - 81 + 27

$54 - 81 + 27$

Step 1.4.2.3.8

Subtract

81

$81$ from

54

$54$ .

- 27 + 27

$- 27 + 27$

Step 1.4.2.3.9

Add

- 27

$- 27$ and

27

$27$ .

0

$0$

0

$0$

Step 1.4.2.4

Since

3

$3$ is a known root, divide the polynomial by

y - 3

$y - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

\frac{- y^{3} + 9 y^{2} - 27 y + 27}{y - 3}

$\frac{- y^{3} + 9 y^{2} - 27 y + 27}{y - 3}$

Step 1.4.2.5

Divide

- y^{3} + 9 y^{2} - 27 y + 27

$- y^{3} + 9 y^{2} - 27 y + 27$ by

y - 3

$y - 3$ .

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Step 1.4.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of

0

$0$ .

y

$y$

3

$3$

y^{3}

$y^{3}$

9 y^{2}

$9 y^{2}$

27 y

$27 y$

27

$27$

Step 1.4.2.5.2

Divide the highest order term in the dividend

- y^{3}

$- y^{3}$ by the highest order term in divisor

y

$y$ .

				-	$y^{2}$ $y^{2}$
	$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$

Step 1.4.2.5.3

Multiply the new quotient term by the divisor.

			-	$y^{2}$ $y^{2}$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			-	$y^{3}$ $y^{3}$	+	$3 y^{2}$ $3 y^{2}$

Step 1.4.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

- y^{3} + 3 y^{2}

$- y^{3} + 3 y^{2}$

			-	$y^{2}$ $y^{2}$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$

Step 1.4.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$y^{2}$ $y^{2}$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$

Step 1.4.2.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$y^{2}$ $y^{2}$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$

Step 1.4.2.5.7

Divide the highest order term in the dividend

6 y^{2}

$6 y^{2}$ by the highest order term in divisor

y

$y$ .

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$

Step 1.4.2.5.8

Multiply the new quotient term by the divisor.

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					+	$6 y^{2}$ $6 y^{2}$	-	$18 y$ $18 y$

Step 1.4.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

6 y^{2} - 18 y

$6 y^{2} - 18 y$

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$

Step 1.4.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$

Step 1.4.2.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$	+	$27$ $27$

Step 1.4.2.5.12

Divide the highest order term in the dividend

- 9 y

$- 9 y$ by the highest order term in divisor

y

$y$ .

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$	-	$9$ $9$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$	+	$27$ $27$

Step 1.4.2.5.13

Multiply the new quotient term by the divisor.

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$	-	$9$ $9$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$	+	$27$ $27$
							-	$9 y$ $9 y$	+	$27$ $27$

Step 1.4.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

- 9 y + 27

$- 9 y + 27$

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$	-	$9$ $9$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$	+	$27$ $27$
							+	$9 y$ $9 y$	-	$27$ $27$

Step 1.4.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$y^{2}$ $y^{2}$	+	$6 y$ $6 y$	-	$9$ $9$
$y$ $y$	-	$3$ $3$	-	$y^{3}$ $y^{3}$	+	$9 y^{2}$ $9 y^{2}$	-	$27 y$ $27 y$	+	$27$ $27$
			+	$y^{3}$ $y^{3}$	-	$3 y^{2}$ $3 y^{2}$
					+	$6 y^{2}$ $6 y^{2}$	-	$27 y$ $27 y$
					-	$6 y^{2}$ $6 y^{2}$	+	$18 y$ $18 y$
							-	$9 y$ $9 y$	+	$27$ $27$
							+	$9 y$ $9 y$	-	$27$ $27$
										$0$ $0$

Step 1.4.2.5.16

Since the remander is

0

$0$ , the final answer is the quotient.

- y^{2} + 6 y - 9

$- y^{2} + 6 y - 9$

- y^{2} + 6 y - 9

$- y^{2} + 6 y - 9$

Step 1.4.2.6

Write

- y^{3} + 9 y^{2} - 27 y + 27

$- y^{3} + 9 y^{2} - 27 y + 27$ as a set of factors.

(y - 3) (- y^{2} + 6 y - 9) = 0

$(y - 3) (- y^{2} + 6 y - 9) = 0$

(y - 3) (- y^{2} + 6 y - 9) = 0

$(y - 3) (- y^{2} + 6 y - 9) = 0$

Step 1.4.3

Factor.

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Step 1.4.3.1

Factor by grouping.

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Step 1.4.3.1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 1 \cdot - 9 = 9

$a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is

b = 6

$b = 6$ .

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Step 1.4.3.1.1.1

Factor

6

$6$ out of

6 y

$6 y$ .

(y - 3) (- y^{2} + 6 (y) - 9) = 0

$(y - 3) (- y^{2} + 6 (y) - 9) = 0$

Step 1.4.3.1.1.2

Rewrite

6

$6$ as

3

$3$ plus

3

$3$

(y - 3) (- y^{2} + (3 + 3) y - 9) = 0

$(y - 3) (- y^{2} + (3 + 3) y - 9) = 0$

Step 1.4.3.1.1.3

Apply the distributive property.

(y - 3) (- y^{2} + 3 y + 3 y - 9) = 0

$(y - 3) (- y^{2} + 3 y + 3 y - 9) = 0$

(y - 3) (- y^{2} + 3 y + 3 y - 9) = 0

$(y - 3) (- y^{2} + 3 y + 3 y - 9) = 0$

Step 1.4.3.1.2

Factor out the greatest common factor from each group.

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Step 1.4.3.1.2.1

Group the first two terms and the last two terms.

(y - 3) ((- y^{2} + 3 y) + 3 y - 9) = 0

$(y - 3) ((- y^{2} + 3 y) + 3 y - 9) = 0$

Step 1.4.3.1.2.2

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

(y - 3) (y (- y + 3) - 3 (- y + 3)) = 0

$(y - 3) (y (- y + 3) - 3 (- y + 3)) = 0$

(y - 3) (y (- y + 3) - 3 (- y + 3)) = 0

$(y - 3) (y (- y + 3) - 3 (- y + 3)) = 0$

Step 1.4.3.1.3

Factor the polynomial by factoring out the greatest common factor,

- y + 3

$- y + 3$ .

(y - 3) ((- y + 3) (y - 3)) = 0

$(y - 3) ((- y + 3) (y - 3)) = 0$

(y - 3) ((- y + 3) (y - 3)) = 0

$(y - 3) ((- y + 3) (y - 3)) = 0$

Step 1.4.3.2

Remove unnecessary parentheses.

(y - 3) (- y + 3) (y - 3) = 0

$(y - 3) (- y + 3) (y - 3) = 0$

(y - 3) (- y + 3) (y - 3) = 0

$(y - 3) (- y + 3) (y - 3) = 0$

Step 1.4.4

Combine exponents.

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Step 1.4.4.1

Factor

- 1

$- 1$ out of

y

$y$ .

(- 1 (- y) - 3) (- y + 3) (y - 3) = 0

$(- 1 (- y) - 3) (- y + 3) (y - 3) = 0$

Step 1.4.4.2

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

(- 1 (- y) - 1 \cdot 3) (- y + 3) (y - 3) = 0

$(- 1 (- y) - 1 \cdot 3) (- y + 3) (y - 3) = 0$

Step 1.4.4.3

Factor

- 1

$- 1$ out of

- 1 (- y) - 1 (3)

$- 1 (- y) - 1 (3)$ .

- 1 (- y + 3) (- y + 3) (y - 3) = 0

$- 1 (- y + 3) (- y + 3) (y - 3) = 0$

Step 1.4.4.4

Raise

- y + 3

$- y + 3$ to the power of

1

$1$ .

- 1 ((- y + 3) (- y + 3)) (y - 3) = 0

$- 1 ((- y + 3) (- y + 3)) (y - 3) = 0$

Step 1.4.4.5

Raise

- y + 3

$- y + 3$ to the power of

1

$1$ .

- 1 ((- y + 3) (- y + 3)) (y - 3) = 0

$- 1 ((- y + 3) (- y + 3)) (y - 3) = 0$

Step 1.4.4.6

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.4.4.7

Add

1

$1$ and

1

$1$ .

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

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

Step 1.4.4.8

Factor

- 1

$- 1$ out of

- y

$- y$ .

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

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

Step 1.4.4.9

Rewrite

3

$3$ as

- 1 (- 3)

$- 1 (- 3)$ .

- 1 {(- (y) - 1 \cdot - 3)}^{2} (y - 3) = 0

$- 1 {(- (y) - 1 \cdot - 3)}^{2} (y - 3) = 0$

Step 1.4.4.10

Factor

- 1

$- 1$ out of

- (y) - 1 (- 3)

$- (y) - 1 (- 3)$ .

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

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

Step 1.4.4.11

Rewrite

- (y - 3)

$- (y - 3)$ as

- 1 (y - 3)

$- 1 (y - 3)$ .

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

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

Step 1.4.4.12

Apply the product rule to

- 1 (y - 3)

$- 1 (y - 3)$ .

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

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

Step 1.4.4.13

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 1.4.4.14

Multiply

{(y - 3)}^{2}

${(y - 3)}^{2}$ by

1

$1$ .

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

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

Step 1.4.4.15

Raise

y - 3

$y - 3$ to the power of

1

$1$ .

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

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

Step 1.4.4.16

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.4.4.17

Add

1

$1$ and

2

$2$ .

- 1 {(y - 3)}^{3} = 0

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

- 1 {(y - 3)}^{3} = 0

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

- 1 {(y - 3)}^{3} = 0

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

Step 1.5

Divide each term in

- 1 {(y - 3)}^{3} = 0

$- 1 {(y - 3)}^{3} = 0$ by

- 1

$- 1$ and simplify.

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Step 1.5.1

Divide each term in

- 1 {(y - 3)}^{3} = 0

$- 1 {(y - 3)}^{3} = 0$ by

- 1

$- 1$ .

\frac{- 1 {(y - 3)}^{3}}{- 1} = \frac{0}{- 1}

$\frac{- 1 {(y - 3)}^{3}}{- 1} = \frac{0}{- 1}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Dividing two negative values results in a positive value.

\frac{{(y - 3)}^{3}}{1} = \frac{0}{- 1}

$\frac{{(y - 3)}^{3}}{1} = \frac{0}{- 1}$

Step 1.5.2.2

Divide

{(y - 3)}^{3}

${(y - 3)}^{3}$ by

1

$1$ .

{(y - 3)}^{3} = \frac{0}{- 1}

${(y - 3)}^{3} = \frac{0}{- 1}$

{(y - 3)}^{3} = \frac{0}{- 1}

${(y - 3)}^{3} = \frac{0}{- 1}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide

0

$0$ by

- 1

$- 1$ .

{(y - 3)}^{3} = 0

${(y - 3)}^{3} = 0$

{(y - 3)}^{3} = 0

${(y - 3)}^{3} = 0$

{(y - 3)}^{3} = 0

${(y - 3)}^{3} = 0$

Step 1.6

Set the

y - 3

$y - 3$ equal to

0

$0$ .

y - 3 = 0

$y - 3 = 0$

Step 1.7

Add

3

$3$ to both sides of the equation.

y = 3

$y = 3$

y = 3

$y = 3$

Step 2

Set

9 y^{2} - 27 y

$9 y^{2} - 27 y$ equal to

0

$0$ .

3 = 0

$3 = 0$

Step 3

Since

3 \neq 0

$3 \neq 0$ , there are no solutions.

No solution