Algebra Examples

$5 {(x + 4)}^{2} = 2 {(x + 1)}^{2} - 27$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $2 {(x + 1)}^{2} - 27$ .

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Step 1.1.1.1

Simplify each term.

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Step 1.1.1.1.1

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

$5 {(x + 4)}^{2} = 2 ((x + 1) (x + 1)) - 27$

Step 1.1.1.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.1.1.1.2.1

Apply the distributive property.

$5 {(x + 4)}^{2} = 2 (x (x + 1) + 1 (x + 1)) - 27$

Step 1.1.1.1.2.2

Apply the distributive property.

$5 {(x + 4)}^{2} = 2 (x \cdot x + x \cdot 1 + 1 (x + 1)) - 27$

Step 1.1.1.1.2.3

Apply the distributive property.

$5 {(x + 4)}^{2} = 2 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) - 27$

Step 1.1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.1.3.1

Simplify each term.

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Step 1.1.1.1.3.1.1

Multiply $x$ by $x$ .

$5 {(x + 4)}^{2} = 2 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) - 27$

Step 1.1.1.1.3.1.2

Multiply $x$ by $1$ .

$5 {(x + 4)}^{2} = 2 (x^{2} + x + 1 x + 1 \cdot 1) - 27$

Step 1.1.1.1.3.1.3

Multiply $x$ by $1$ .

$5 {(x + 4)}^{2} = 2 (x^{2} + x + x + 1 \cdot 1) - 27$

Step 1.1.1.1.3.1.4

Multiply $1$ by $1$ .

$5 {(x + 4)}^{2} = 2 (x^{2} + x + x + 1) - 27$

Step 1.1.1.1.3.2

Add $x$ and $x$ .

$5 {(x + 4)}^{2} = 2 (x^{2} + 2 x + 1) - 27$

Step 1.1.1.1.4

Apply the distributive property.

$5 {(x + 4)}^{2} = 2 x^{2} + 2 (2 x) + 2 \cdot 1 - 27$

Step 1.1.1.1.5

Simplify.

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Step 1.1.1.1.5.1

Multiply $2$ by $2$ .

$5 {(x + 4)}^{2} = 2 x^{2} + 4 x + 2 \cdot 1 - 27$

Step 1.1.1.1.5.2

Multiply $2$ by $1$ .

$5 {(x + 4)}^{2} = 2 x^{2} + 4 x + 2 - 27$

Step 1.1.1.2

Subtract $27$ from $2$ .

$5 {(x + 4)}^{2} = 2 x^{2} + 4 x - 25$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $2 x^{2}$ from both sides of the equation.

$5 {(x + 4)}^{2} - 2 x^{2} = 4 x - 25$

Step 1.2.2

Subtract $4 x$ from both sides of the equation.

$5 {(x + 4)}^{2} - 2 x^{2} - 4 x = - 25$

Step 1.2.3

Add $25$ to both sides of the equation.

$5 {(x + 4)}^{2} - 2 x^{2} - 4 x + 25 = 0$

Step 1.3

Simplify $5 {(x + 4)}^{2} - 2 x^{2} - 4 x + 25$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$5 ((x + 4) (x + 4)) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Apply the distributive property.

$5 (x (x + 4) + 4 (x + 4)) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.2.2

Apply the distributive property.

$5 (x \cdot x + x \cdot 4 + 4 (x + 4)) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.2.3

Apply the distributive property.

$5 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $x$ by $x$ .

$5 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.3.1.2

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

$5 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.3.1.3

Multiply $4$ by $4$ .

$5 (x^{2} + 4 x + 4 x + 16) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.3.2

Add $4 x$ and $4 x$ .

$5 (x^{2} + 8 x + 16) - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.4

Apply the distributive property.

$5 x^{2} + 5 (8 x) + 5 \cdot 16 - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.5

Simplify.

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Step 1.3.1.5.1

Multiply $8$ by $5$ .

$5 x^{2} + 40 x + 5 \cdot 16 - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.1.5.2

Multiply $5$ by $16$ .

$5 x^{2} + 40 x + 80 - 2 x^{2} - 4 x + 25 = 0$

Step 1.3.2

Subtract $2 x^{2}$ from $5 x^{2}$ .

$3 x^{2} + 40 x + 80 - 4 x + 25 = 0$

Step 1.3.3

Subtract $4 x$ from $40 x$ .

$3 x^{2} + 36 x + 80 + 25 = 0$

Step 1.3.4

Add $80$ and $25$ .

$3 x^{2} + 36 x + 105 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 3$ , $b = 36$ , and $c = 105$ into the quadratic formula and solve for $x$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (3 \cdot 105)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $36$ to the power of $2$ .

$x = \frac{- 36 \pm \sqrt{1296 - 4 \cdot 3 \cdot 105}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 105$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 36 \pm \sqrt{1296 - 12 \cdot 105}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $105$ .

$x = \frac{- 36 \pm \sqrt{1296 - 1260}}{2 \cdot 3}$

Step 4.1.3

Subtract $1260$ from $1296$ .

$x = \frac{- 36 \pm \sqrt{36}}{2 \cdot 3}$

Step 4.1.4

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 36 \pm \sqrt{6^{2}}}{2 \cdot 3}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 36 \pm 6}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{- 36 \pm 6}{6}$

Step 4.3

Simplify $\frac{- 36 \pm 6}{6}$ .

$x = - 6 \pm 1$

Step 5

The final answer is the combination of both solutions.

$x = - 5, - 7$