Pre-Algebra Examples

$5.2 x + 9.8 x (178 x) = 1358$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$5.2 x + 9.8 \cdot 178 x \cdot x = 1358$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$5.2 x + 9.8 \cdot 178 (x \cdot x) = 1358$

Step 1.2.2

Multiply $x$ by $x$ .

$5.2 x + 9.8 \cdot 178 x^{2} = 1358$

Step 1.3

Multiply $9.8$ by $178$ .

$5.2 x + 1744.4 x^{2} = 1358$

Step 2

Subtract $1358$ from both sides of the equation.

$5.2 x + 1744.4 x^{2} - 1358 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $0.4$ out of $5.2 x + 1744.4 x^{2} - 1358$ .

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Step 3.1.1

Factor $0.4$ out of $5.2 x$ .

$0.4 (13 x) + 1744.4 x^{2} - 1358 = 0$

Step 3.1.2

Factor $0.4$ out of $1744.4 x^{2}$ .

$0.4 (13 x) + 0.4 (4361 x^{2}) - 1358 = 0$

Step 3.1.3

Factor $0.4$ out of $- 1358$ .

$0.4 (13 x) + 0.4 (4361 x^{2}) + 0.4 (- 3395) = 0$

Step 3.1.4

Factor $0.4$ out of $0.4 (13 x) + 0.4 (4361 x^{2})$ .

$0.4 (13 x + 4361 x^{2}) + 0.4 (- 3395) = 0$

Step 3.1.5

Factor $0.4$ out of $0.4 (13 x + 4361 x^{2}) + 0.4 (- 3395)$ .

$0.4 (13 x + 4361 x^{2} - 3395) = 0$

Step 3.2

Reorder terms.

$0.4 (4361 x^{2} + 13 x - 3395) = 0$

Step 4

Divide each term in $0.4 (4361 x^{2} + 13 x - 3395) = 0$ by $0.4$ and simplify.

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Step 4.1

Divide each term in $0.4 (4361 x^{2} + 13 x - 3395) = 0$ by $0.4$ .

$\frac{0.4 (4361 x^{2} + 13 x - 3395)}{0.4} = \frac{0}{0.4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.4 (4361 x^{2} + 13 x - 3395)}{0.4} = \frac{0}{0.4}$

Step 4.2.1.2

Divide $4361 x^{2} + 13 x - 3395$ by $1$ .

$4361 x^{2} + 13 x - 3395 = \frac{0}{0.4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.4$ .

$4361 x^{2} + 13 x - 3395 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 4361$ , $b = 13$ , and $c = - 3395$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (4361 \cdot - 3395)}}{2 \cdot 4361}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 4361 \cdot - 3395}}{2 \cdot 4361}$

Step 7.1.2

Multiply $- 4 \cdot 4361 \cdot - 3395$ .

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Step 7.1.2.1

Multiply $- 4$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{169 - 17444 \cdot - 3395}}{2 \cdot 4361}$

Step 7.1.2.2

Multiply $- 17444$ by $- 3395$ .

$x = \frac{- 13 \pm \sqrt{169 + 59222380}}{2 \cdot 4361}$

Step 7.1.3

Add $169$ and $59222380$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{2 \cdot 4361}$

Step 7.2

Multiply $2$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{8722}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 4361 \cdot - 3395}}{2 \cdot 4361}$

Step 8.1.2

Multiply $- 4 \cdot 4361 \cdot - 3395$ .

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Step 8.1.2.1

Multiply $- 4$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{169 - 17444 \cdot - 3395}}{2 \cdot 4361}$

Step 8.1.2.2

Multiply $- 17444$ by $- 3395$ .

$x = \frac{- 13 \pm \sqrt{169 + 59222380}}{2 \cdot 4361}$

Step 8.1.3

Add $169$ and $59222380$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{2 \cdot 4361}$

Step 8.2

Multiply $2$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{8722}$

Step 8.3

Change the $\pm$ to $+$ .

$x = \frac{- 13 + \sqrt{59222549}}{8722}$

Step 8.4

Rewrite $- 13$ as $- 1 (13)$ .

$x = \frac{- 1 \cdot 13 + \sqrt{59222549}}{8722}$

Step 8.5

Factor $- 1$ out of $\sqrt{59222549}$ .

$x = \frac{- 1 \cdot 13 - 1 (- \sqrt{59222549})}{8722}$

Step 8.6

Factor $- 1$ out of $- 1 (13) - 1 (- \sqrt{59222549})$ .

$x = \frac{- 1 (13 - \sqrt{59222549})}{8722}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{13 - \sqrt{59222549}}{8722}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 4361 \cdot - 3395}}{2 \cdot 4361}$

Step 9.1.2

Multiply $- 4 \cdot 4361 \cdot - 3395$ .

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Step 9.1.2.1

Multiply $- 4$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{169 - 17444 \cdot - 3395}}{2 \cdot 4361}$

Step 9.1.2.2

Multiply $- 17444$ by $- 3395$ .

$x = \frac{- 13 \pm \sqrt{169 + 59222380}}{2 \cdot 4361}$

Step 9.1.3

Add $169$ and $59222380$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{2 \cdot 4361}$

Step 9.2

Multiply $2$ by $4361$ .

$x = \frac{- 13 \pm \sqrt{59222549}}{8722}$

Step 9.3

Change the $\pm$ to $-$ .

$x = \frac{- 13 - \sqrt{59222549}}{8722}$

Step 9.4

Rewrite $- 13$ as $- 1 (13)$ .

$x = \frac{- 1 \cdot 13 - \sqrt{59222549}}{8722}$

Step 9.5

Factor $- 1$ out of $- \sqrt{59222549}$ .

$x = \frac{- 1 \cdot 13 - (\sqrt{59222549})}{8722}$

Step 9.6

Factor $- 1$ out of $- 1 (13) - (\sqrt{59222549})$ .

$x = \frac{- 1 (13 + \sqrt{59222549})}{8722}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{13 + \sqrt{59222549}}{8722}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{13 - \sqrt{59222549}}{8722}, - \frac{13 + \sqrt{59222549}}{8722}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{13 - \sqrt{59222549}}{8722}, - \frac{13 + \sqrt{59222549}}{8722}$

Decimal Form:

$x = 0.88083224 \dots, - 0.88381321 \dots$