Linear Algebra Examples

$5 {(s + 8)}^{2} - 19 (s + 8) = 4$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$5 {(s + 8)}^{2} - 19 s - 19 \cdot 8 = 4$

Step 1.2

Multiply $- 19$ by $8$ .

$5 {(s + 8)}^{2} - 19 s - 152 = 4$

Step 2

Subtract $4$ from both sides of the equation.

$5 {(s + 8)}^{2} - 19 s - 152 - 4 = 0$

Step 3

Simplify $5 {(s + 8)}^{2} - 19 s - 152 - 4$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(s + 8)}^{2}$ as $(s + 8) (s + 8)$ .

$5 ((s + 8) (s + 8)) - 19 s - 152 - 4 = 0$

Step 3.1.2

Expand $(s + 8) (s + 8)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$5 (s (s + 8) + 8 (s + 8)) - 19 s - 152 - 4 = 0$

Step 3.1.2.2

Apply the distributive property.

$5 (s \cdot s + s \cdot 8 + 8 (s + 8)) - 19 s - 152 - 4 = 0$

Step 3.1.2.3

Apply the distributive property.

$5 (s \cdot s + s \cdot 8 + 8 s + 8 \cdot 8) - 19 s - 152 - 4 = 0$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $s$ by $s$ .

$5 (s^{2} + s \cdot 8 + 8 s + 8 \cdot 8) - 19 s - 152 - 4 = 0$

Step 3.1.3.1.2

Move $8$ to the left of $s$ .

$5 (s^{2} + 8 \cdot s + 8 s + 8 \cdot 8) - 19 s - 152 - 4 = 0$

Step 3.1.3.1.3

Multiply $8$ by $8$ .

$5 (s^{2} + 8 s + 8 s + 64) - 19 s - 152 - 4 = 0$

Step 3.1.3.2

Add $8 s$ and $8 s$ .

$5 (s^{2} + 16 s + 64) - 19 s - 152 - 4 = 0$

Step 3.1.4

Apply the distributive property.

$5 s^{2} + 5 (16 s) + 5 \cdot 64 - 19 s - 152 - 4 = 0$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $16$ by $5$ .

$5 s^{2} + 80 s + 5 \cdot 64 - 19 s - 152 - 4 = 0$

Step 3.1.5.2

Multiply $5$ by $64$ .

$5 s^{2} + 80 s + 320 - 19 s - 152 - 4 = 0$

Step 3.2

Subtract $19 s$ from $80 s$ .

$5 s^{2} + 61 s + 320 - 152 - 4 = 0$

Step 3.3

Subtract $152$ from $320$ .

$5 s^{2} + 61 s + 168 - 4 = 0$

Step 3.4

Subtract $4$ from $168$ .

$5 s^{2} + 61 s + 164 = 0$

Step 4

Factor by grouping.

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Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 164 = 820$ and whose sum is $b = 61$ .

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Step 4.1.1

Factor $61$ out of $61 s$ .

$5 s^{2} + 61 (s) + 164 = 0$

Step 4.1.2

Rewrite $61$ as $20$ plus $41$

$5 s^{2} + (20 + 41) s + 164 = 0$

Step 4.1.3

Apply the distributive property.

$5 s^{2} + 20 s + 41 s + 164 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$(5 s^{2} + 20 s) + 41 s + 164 = 0$

Step 4.2.2

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

$5 s (s + 4) + 41 (s + 4) = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $s + 4$ .

$(s + 4) (5 s + 41) = 0$

Step 5

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

$s + 4 = 0$

$5 s + 41 = 0$

Step 6

Set $s + 4$ equal to $0$ and solve for $s$ .

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Step 6.1

Set $s + 4$ equal to $0$ .

$s + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$s = - 4$

Step 7

Set $5 s + 41$ equal to $0$ and solve for $s$ .

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Step 7.1

Set $5 s + 41$ equal to $0$ .

$5 s + 41 = 0$

Step 7.2

Solve $5 s + 41 = 0$ for $s$ .

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Step 7.2.1

Subtract $41$ from both sides of the equation.

$5 s = - 41$

Step 7.2.2

Divide each term in $5 s = - 41$ by $5$ and simplify.

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Step 7.2.2.1

Divide each term in $5 s = - 41$ by $5$ .

$\frac{5 s}{5} = \frac{- 41}{5}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $5$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{5 s}{5} = \frac{- 41}{5}$

Step 7.2.2.2.1.2

Divide $s$ by $1$ .

$s = \frac{- 41}{5}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$s = - \frac{41}{5}$

Step 8

The final solution is all the values that make $(s + 4) (5 s + 41) = 0$ true.

$s = - 4, - \frac{41}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$s = - 4, - \frac{41}{5}$

Decimal Form:

$s = - 4, - 8.2$

Mixed Number Form:

$s = - 4, - 8 \frac{1}{5}$