Pre-Algebra Examples

$47 x + 111 + 44 x (33 x + 1) = 360$

Step 1

Simplify $47 x + 111 + 44 x (33 x + 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$47 x + 111 + 44 x (33 x) + 44 x \cdot 1 = 360$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$47 x + 111 + 44 \cdot 33 x \cdot x + 44 x \cdot 1 = 360$

Step 1.1.3

Multiply $44$ by $1$ .

$47 x + 111 + 44 \cdot 33 x \cdot x + 44 x = 360$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

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

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Step 1.1.4.1.1

Move $x$ .

$47 x + 111 + 44 \cdot 33 (x \cdot x) + 44 x = 360$

Step 1.1.4.1.2

Multiply $x$ by $x$ .

$47 x + 111 + 44 \cdot 33 x^{2} + 44 x = 360$

Step 1.1.4.2

Multiply $44$ by $33$ .

$47 x + 111 + 1452 x^{2} + 44 x = 360$

Step 1.2

Add $47 x$ and $44 x$ .

$91 x + 111 + 1452 x^{2} = 360$

Step 2

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

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Step 2.1

Subtract $360$ from both sides of the equation.

$91 x + 111 + 1452 x^{2} - 360 = 0$

Step 2.2

Subtract $360$ from $111$ .

$91 x + 1452 x^{2} - 249 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1452$ , $b = 91$ , and $c = - 249$ into the quadratic formula and solve for $x$ .

$\frac{- 91 \pm \sqrt{91^{2} - 4 \cdot (1452 \cdot - 249)}}{2 \cdot 1452}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 1452 \cdot - 249}}{2 \cdot 1452}$

Step 5.1.2

Multiply $- 4 \cdot 1452 \cdot - 249$ .

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Step 5.1.2.1

Multiply $- 4$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{8281 - 5808 \cdot - 249}}{2 \cdot 1452}$

Step 5.1.2.2

Multiply $- 5808$ by $- 249$ .

$x = \frac{- 91 \pm \sqrt{8281 + 1446192}}{2 \cdot 1452}$

Step 5.1.3

Add $8281$ and $1446192$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2 \cdot 1452}$

Step 5.2

Multiply $2$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2904}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 1452 \cdot - 249}}{2 \cdot 1452}$

Step 6.1.2

Multiply $- 4 \cdot 1452 \cdot - 249$ .

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Step 6.1.2.1

Multiply $- 4$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{8281 - 5808 \cdot - 249}}{2 \cdot 1452}$

Step 6.1.2.2

Multiply $- 5808$ by $- 249$ .

$x = \frac{- 91 \pm \sqrt{8281 + 1446192}}{2 \cdot 1452}$

Step 6.1.3

Add $8281$ and $1446192$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2 \cdot 1452}$

Step 6.2

Multiply $2$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2904}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 91 + \sqrt{1454473}}{2904}$

Step 6.4

Rewrite $- 91$ as $- 1 (91)$ .

$x = \frac{- 1 \cdot 91 + \sqrt{1454473}}{2904}$

Step 6.5

Factor $- 1$ out of $\sqrt{1454473}$ .

$x = \frac{- 1 \cdot 91 - 1 (- \sqrt{1454473})}{2904}$

Step 6.6

Factor $- 1$ out of $- 1 (91) - 1 (- \sqrt{1454473})$ .

$x = \frac{- 1 (91 - \sqrt{1454473})}{2904}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{91 - \sqrt{1454473}}{2904}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 1452 \cdot - 249}}{2 \cdot 1452}$

Step 7.1.2

Multiply $- 4 \cdot 1452 \cdot - 249$ .

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Step 7.1.2.1

Multiply $- 4$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{8281 - 5808 \cdot - 249}}{2 \cdot 1452}$

Step 7.1.2.2

Multiply $- 5808$ by $- 249$ .

$x = \frac{- 91 \pm \sqrt{8281 + 1446192}}{2 \cdot 1452}$

Step 7.1.3

Add $8281$ and $1446192$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2 \cdot 1452}$

Step 7.2

Multiply $2$ by $1452$ .

$x = \frac{- 91 \pm \sqrt{1454473}}{2904}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 91 - \sqrt{1454473}}{2904}$

Step 7.4

Rewrite $- 91$ as $- 1 (91)$ .

$x = \frac{- 1 \cdot 91 - \sqrt{1454473}}{2904}$

Step 7.5

Factor $- 1$ out of $- \sqrt{1454473}$ .

$x = \frac{- 1 \cdot 91 - (\sqrt{1454473})}{2904}$

Step 7.6

Factor $- 1$ out of $- 1 (91) - (\sqrt{1454473})$ .

$x = \frac{- 1 (91 + \sqrt{1454473})}{2904}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{91 + \sqrt{1454473}}{2904}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{91 - \sqrt{1454473}}{2904}, - \frac{91 + \sqrt{1454473}}{2904}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{91 - \sqrt{1454473}}{2904}, - \frac{91 + \sqrt{1454473}}{2904}$

Decimal Form:

$x = 0.38395845 \dots, - 0.44663062 \dots$