Algebra Examples

${(2 y + 1)}^{2} - 4 y^{2} = 5$

Step 1

Subtract $5$ from both sides of the equation.

${(2 y + 1)}^{2} - 4 y^{2} - 5 = 0$

Step 2

Simplify ${(2 y + 1)}^{2} - 4 y^{2} - 5$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(2 y + 1)}^{2}$ as $(2 y + 1) (2 y + 1)$ .

$(2 y + 1) (2 y + 1) - 4 y^{2} - 5 = 0$

Step 2.1.2

Expand $(2 y + 1) (2 y + 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$2 y (2 y + 1) + 1 (2 y + 1) - 4 y^{2} - 5 = 0$

Step 2.1.2.2

Apply the distributive property.

$2 y (2 y) + 2 y \cdot 1 + 1 (2 y + 1) - 4 y^{2} - 5 = 0$

Step 2.1.2.3

Apply the distributive property.

$2 y (2 y) + 2 y \cdot 1 + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 y \cdot y + 2 y \cdot 1 + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.3.1.2.1

Move $y$ .

$2 \cdot 2 (y \cdot y) + 2 y \cdot 1 + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.2.2

Multiply $y$ by $y$ .

$2 \cdot 2 y^{2} + 2 y \cdot 1 + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$4 y^{2} + 2 y \cdot 1 + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.4

Multiply $2$ by $1$ .

$4 y^{2} + 2 y + 1 (2 y) + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.5

Multiply $2 y$ by $1$ .

$4 y^{2} + 2 y + 2 y + 1 \cdot 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.1.6

Multiply $1$ by $1$ .

$4 y^{2} + 2 y + 2 y + 1 - 4 y^{2} - 5 = 0$

Step 2.1.3.2

Add $2 y$ and $2 y$ .

$4 y^{2} + 4 y + 1 - 4 y^{2} - 5 = 0$

Step 2.2

Combine the opposite terms in $4 y^{2} + 4 y + 1 - 4 y^{2} - 5$ .

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Step 2.2.1

Subtract $4 y^{2}$ from $4 y^{2}$ .

$4 y + 1 + 0 - 5 = 0$

Step 2.2.2

Add $4 y + 1$ and $0$ .

$4 y + 1 - 5 = 0$

Step 2.3

Subtract $5$ from $1$ .

$4 y - 4 = 0$

Step 3

Factor $4$ out of $4 y - 4$ .

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Step 3.1

Factor $4$ out of $4 y$ .

$4 (y) - 4 = 0$

Step 3.2

Factor $4$ out of $- 4$ .

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

Step 3.3

Factor $4$ out of $4 (y) + 4 (- 1)$ .

$4 (y - 1) = 0$

Step 4

Divide each term in $4 (y - 1) = 0$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 (y - 1) = 0$ by $4$ .

$\frac{4 (y - 1)}{4} = \frac{0}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 (y - 1)}{4} = \frac{0}{4}$

Step 4.2.1.2

Divide $y - 1$ by $1$ .

$y - 1 = \frac{0}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $4$ .

$y - 1 = 0$

Step 5

Add $1$ to both sides of the equation.

$y = 1$