Finite Math Examples

${(8 x + 2)}^{2} = 45$

Step 1

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

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Step 1.1

Subtract $45$ from both sides of the equation.

${(8 x + 2)}^{2} - 45 = 0$

Step 1.2

Simplify ${(8 x + 2)}^{2} - 45$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$(8 x + 2) (8 x + 2) - 45 = 0$

Step 1.2.1.2

Expand $(8 x + 2) (8 x + 2)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$8 x (8 x + 2) + 2 (8 x + 2) - 45 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$8 x (8 x) + 8 x \cdot 2 + 2 (8 x + 2) - 45 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$8 x (8 x) + 8 x \cdot 2 + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$8 \cdot 8 x \cdot x + 8 x \cdot 2 + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.2

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

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Step 1.2.1.3.1.2.1

Move $x$ .

$8 \cdot 8 (x \cdot x) + 8 x \cdot 2 + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.2.2

Multiply $x$ by $x$ .

$8 \cdot 8 x^{2} + 8 x \cdot 2 + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.3

Multiply $8$ by $8$ .

$64 x^{2} + 8 x \cdot 2 + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.4

Multiply $2$ by $8$ .

$64 x^{2} + 16 x + 2 (8 x) + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.5

Multiply $8$ by $2$ .

$64 x^{2} + 16 x + 16 x + 2 \cdot 2 - 45 = 0$

Step 1.2.1.3.1.6

Multiply $2$ by $2$ .

$64 x^{2} + 16 x + 16 x + 4 - 45 = 0$

Step 1.2.1.3.2

Add $16 x$ and $16 x$ .

$64 x^{2} + 32 x + 4 - 45 = 0$

Step 1.2.2

Subtract $45$ from $4$ .

$64 x^{2} + 32 x - 41 = 0$

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 $a = 64$ , $b = 32$ , and $c = - 41$ into the quadratic formula and solve for $x$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (64 \cdot - 41)}}{2 \cdot 64}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 64 \cdot - 41}}{2 \cdot 64}$

Step 4.1.2

Multiply $- 4 \cdot 64 \cdot - 41$ .

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Step 4.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{- 32 \pm \sqrt{1024 - 256 \cdot - 41}}{2 \cdot 64}$

Step 4.1.2.2

Multiply $- 256$ by $- 41$ .

$x = \frac{- 32 \pm \sqrt{1024 + 10496}}{2 \cdot 64}$

Step 4.1.3

Add $1024$ and $10496$ .

$x = \frac{- 32 \pm \sqrt{11520}}{2 \cdot 64}$

Step 4.1.4

Rewrite $11520$ as $48^{2} \cdot 5$ .

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Step 4.1.4.1

Factor $2304$ out of $11520$ .

$x = \frac{- 32 \pm \sqrt{2304 (5)}}{2 \cdot 64}$

Step 4.1.4.2

Rewrite $2304$ as $48^{2}$ .

$x = \frac{- 32 \pm \sqrt{48^{2} \cdot 5}}{2 \cdot 64}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 32 \pm 48 \sqrt{5}}{2 \cdot 64}$

Step 4.2

Multiply $2$ by $64$ .

$x = \frac{- 32 \pm 48 \sqrt{5}}{128}$

Step 4.3

Simplify $\frac{- 32 \pm 48 \sqrt{5}}{128}$ .

$x = \frac{- 2 \pm 3 \sqrt{5}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 2 \pm 3 \sqrt{5}}{8}$

Decimal Form:

$x = 0.58852549 \dots, - 1.08852549 \dots$