Algebra Examples

${(x - 3)}^{2} - {(2 x + 5)}^{2} = - 16$

Step 1

Add $16$ to both sides of the equation.

${(x - 3)}^{2} - {(2 x + 5)}^{2} + 16 = 0$

Step 2

Simplify ${(x - 3)}^{2} - {(2 x + 5)}^{2} + 16$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(x - 3) (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x (x - 3) - 3 (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

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

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 - {(2 x + 5)}^{2} + 16 = 0$

Step 2.1.4

Rewrite ${(2 x + 5)}^{2}$ as $(2 x + 5) (2 x + 5)$ .

$x^{2} - 6 x + 9 - ((2 x + 5) (2 x + 5)) + 16 = 0$

Step 2.1.5

Expand $(2 x + 5) (2 x + 5)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x + 5) + 5 (2 x + 5)) + 16 = 0$

Step 2.1.5.2

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5)) + 16 = 0$

Step 2.1.5.3

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

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

Rewrite using the commutative property of multiplication.

$x^{2} - 6 x + 9 - (2 \cdot 2 x \cdot x + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.2

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

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Step 2.1.6.1.2.1

Move $x$ .

$x^{2} - 6 x + 9 - (2 \cdot 2 (x \cdot x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.2.2

Multiply $x$ by $x$ .

$x^{2} - 6 x + 9 - (2 \cdot 2 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.3

Multiply $2$ by $2$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.4

Multiply $5$ by $2$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.5

Multiply $2$ by $5$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 10 x + 5 \cdot 5) + 16 = 0$

Step 2.1.6.1.6

Multiply $5$ by $5$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 10 x + 25) + 16 = 0$

Step 2.1.6.2

Add $10 x$ and $10 x$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 20 x + 25) + 16 = 0$

Step 2.1.7

Apply the distributive property.

$x^{2} - 6 x + 9 - (4 x^{2}) - (20 x) - 1 \cdot 25 + 16 = 0$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $4$ by $- 1$ .

$x^{2} - 6 x + 9 - 4 x^{2} - (20 x) - 1 \cdot 25 + 16 = 0$

Step 2.1.8.2

Multiply $20$ by $- 1$ .

$x^{2} - 6 x + 9 - 4 x^{2} - 20 x - 1 \cdot 25 + 16 = 0$

Step 2.1.8.3

Multiply $- 1$ by $25$ .

$x^{2} - 6 x + 9 - 4 x^{2} - 20 x - 25 + 16 = 0$

Step 2.2

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

$- 3 x^{2} - 6 x + 9 - 20 x - 25 + 16 = 0$

Step 2.3

Subtract $20 x$ from $- 6 x$ .

$- 3 x^{2} - 26 x + 9 - 25 + 16 = 0$

Step 2.4

Subtract $25$ from $9$ .

$- 3 x^{2} - 26 x - 16 + 16 = 0$

Step 2.5

Combine the opposite terms in $- 3 x^{2} - 26 x - 16 + 16$ .

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Step 2.5.1

Add $- 16$ and $16$ .

$- 3 x^{2} - 26 x + 0 = 0$

Step 2.5.2

Add $- 3 x^{2} - 26 x$ and $0$ .

$- 3 x^{2} - 26 x = 0$

Step 3

Factor $- x$ out of $- 3 x^{2} - 26 x$ .

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Step 3.1

Factor $- x$ out of $- 3 x^{2}$ .

$- x (3 x) - 26 x = 0$

Step 3.2

Factor $- x$ out of $- 26 x$ .

$- x (3 x) - x \cdot 26 = 0$

Step 3.3

Factor $- x$ out of $- x (3 x) - x (26)$ .

$- x (3 x + 26) = 0$

Step 4

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

$x = 0$

$3 x + 26 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $3 x + 26$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $3 x + 26$ equal to $0$ .

$3 x + 26 = 0$

Step 6.2

Solve $3 x + 26 = 0$ for $x$ .

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Step 6.2.1

Subtract $26$ from both sides of the equation.

$3 x = - 26$

Step 6.2.2

Divide each term in $3 x = - 26$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 x = - 26$ by $3$ .

$\frac{3 x}{3} = \frac{- 26}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 26}{3}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 26}{3}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{26}{3}$

Step 7

The final solution is all the values that make $- x (3 x + 26) = 0$ true.

$x = 0, - \frac{26}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{26}{3}$

Decimal Form:

$x = 0, - 8.\overline{6}$

Mixed Number Form:

$x = 0, - 8 \frac{2}{3}$