Algebra Examples

${(2 x + 3)}^{2} + {(x + 4)}^{2} = 10$

Step 1

Subtract $10$ from both sides of the equation.

${(2 x + 3)}^{2} + {(x + 4)}^{2} - 10 = 0$

Step 2

Simplify ${(2 x + 3)}^{2} + {(x + 4)}^{2} - 10$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(2 x + 3)}^{2}$ as $(2 x + 3) (2 x + 3)$ .

$(2 x + 3) (2 x + 3) + {(x + 4)}^{2} - 10 = 0$

Step 2.1.2

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$2 x (2 x + 3) + 3 (2 x + 3) + {(x + 4)}^{2} - 10 = 0$

Step 2.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3) + {(x + 4)}^{2} - 10 = 0$

Step 2.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

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.

$2 \cdot 2 x \cdot x + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.4

Multiply $3$ by $2$ .

$4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.5

Multiply $2$ by $3$ .

$4 x^{2} + 6 x + 6 x + 3 \cdot 3 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.1.6

Multiply $3$ by $3$ .

$4 x^{2} + 6 x + 6 x + 9 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.3.2

Add $6 x$ and $6 x$ .

$4 x^{2} + 12 x + 9 + {(x + 4)}^{2} - 10 = 0$

Step 2.1.4

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$4 x^{2} + 12 x + 9 + (x + 4) (x + 4) - 10 = 0$

Step 2.1.5

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$4 x^{2} + 12 x + 9 + x (x + 4) + 4 (x + 4) - 10 = 0$

Step 2.1.5.2

Apply the distributive property.

$4 x^{2} + 12 x + 9 + x \cdot x + x \cdot 4 + 4 (x + 4) - 10 = 0$

Step 2.1.5.3

Apply the distributive property.

$4 x^{2} + 12 x + 9 + x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - 10 = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $x$ by $x$ .

$4 x^{2} + 12 x + 9 + x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - 10 = 0$

Step 2.1.6.1.2

Move $4$ to the left of $x$ .

$4 x^{2} + 12 x + 9 + x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - 10 = 0$

Step 2.1.6.1.3

Multiply $4$ by $4$ .

$4 x^{2} + 12 x + 9 + x^{2} + 4 x + 4 x + 16 - 10 = 0$

Step 2.1.6.2

Add $4 x$ and $4 x$ .

$4 x^{2} + 12 x + 9 + x^{2} + 8 x + 16 - 10 = 0$

Step 2.2

Add $4 x^{2}$ and $x^{2}$ .

$5 x^{2} + 12 x + 9 + 8 x + 16 - 10 = 0$

Step 2.3

Add $12 x$ and $8 x$ .

$5 x^{2} + 20 x + 9 + 16 - 10 = 0$

Step 2.4

Add $9$ and $16$ .

$5 x^{2} + 20 x + 25 - 10 = 0$

Step 2.5

Subtract $10$ from $25$ .

$5 x^{2} + 20 x + 15 = 0$

Step 3

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c = 0$

Step 4

The standard form is $5 x^{2} + 20 x + 15$ .

$5 x^{2} + 20 x + 15 = 0$

Step 5