Pre-Algebra Examples

$7 x - 4 (x - 1) = 9 x^{2} \cdot 1 - 5 x$

Step 1

Simplify $7 x - 4 (x - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$7 x - 4 x - 4 \cdot - 1 = 9 x^{2} \cdot 1 - 5 x$

Step 1.1.2

Multiply $- 4$ by $- 1$ .

$7 x - 4 x + 4 = 9 x^{2} \cdot 1 - 5 x$

Step 1.2

Subtract $4 x$ from $7 x$ .

$3 x + 4 = 9 x^{2} \cdot 1 - 5 x$

Step 2

Multiply $9$ by $1$ .

$3 x + 4 = 9 x^{2} - 5 x$

Step 3

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 x^{2} - 5 x = 3 x + 4$

Step 4

Move all terms containing $x$ to the left side of the equation.

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Step 4.1

Subtract $3 x$ from both sides of the equation.

$9 x^{2} - 5 x - 3 x = 4$

Step 4.2

Subtract $3 x$ from $- 5 x$ .

$9 x^{2} - 8 x = 4$

Step 5

Subtract $4$ from both sides of the equation.

$9 x^{2} - 8 x - 4 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 9$ , $b = - 8$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 8.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 8.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 8.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 8.1.3

Add $64$ and $144$ .

$x = \frac{8 \pm \sqrt{208}}{2 \cdot 9}$

Step 8.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 8.1.4.1

Factor $16$ out of $208$ .

$x = \frac{8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 8.2

Multiply $2$ by $9$ .

$x = \frac{8 \pm 4 \sqrt{13}}{18}$

Step 8.3

Simplify $\frac{8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{4 \pm 2 \sqrt{13}}{9}$

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 $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 9.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 9.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 9.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 9.1.3

Add $64$ and $144$ .

$x = \frac{8 \pm \sqrt{208}}{2 \cdot 9}$

Step 9.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 9.1.4.1

Factor $16$ out of $208$ .

$x = \frac{8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 9.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 9.2

Multiply $2$ by $9$ .

$x = \frac{8 \pm 4 \sqrt{13}}{18}$

Step 9.3

Simplify $\frac{8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{4 \pm 2 \sqrt{13}}{9}$

Step 9.4

Change the $\pm$ to $+$ .

$x = \frac{4 + 2 \sqrt{13}}{9}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 10.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 10.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 10.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 10.1.3

Add $64$ and $144$ .

$x = \frac{8 \pm \sqrt{208}}{2 \cdot 9}$

Step 10.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 10.1.4.1

Factor $16$ out of $208$ .

$x = \frac{8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 10.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 10.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 10.2

Multiply $2$ by $9$ .

$x = \frac{8 \pm 4 \sqrt{13}}{18}$

Step 10.3

Simplify $\frac{8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{4 \pm 2 \sqrt{13}}{9}$

Step 10.4

Change the $\pm$ to $-$ .

$x = \frac{4 - 2 \sqrt{13}}{9}$

Step 11

The final answer is the combination of both solutions.

$x = \frac{4 + 2 \sqrt{13}}{9}, \frac{4 - 2 \sqrt{13}}{9}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 + 2 \sqrt{13}}{9}, \frac{4 - 2 \sqrt{13}}{9}$

Decimal Form:

$x = 1.24567806 \dots, - 0.35678917 \dots$