Pre-Algebra Examples

$x + 4 x + 3 - x \cdot x - 2 = 48$

Step 1

Simplify $x + 4 x + 3 - x \cdot x - 2$ .

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Step 1.1

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

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Step 1.1.1

Move $x$ .

$x + 4 x + 3 - (x \cdot x) - 2 = 48$

Step 1.1.2

Multiply $x$ by $x$ .

$x + 4 x + 3 - x^{2} - 2 = 48$

Step 1.2

Add $x$ and $4 x$ .

$5 x + 3 - x^{2} - 2 = 48$

Step 1.3

Subtract $2$ from $3$ .

$5 x - x^{2} + 1 = 48$

Step 2

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

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Step 2.1

Subtract $48$ from both sides of the equation.

$5 x - x^{2} + 1 - 48 = 0$

Step 2.2

Subtract $48$ from $1$ .

$5 x - x^{2} - 47 = 0$

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 $a = - 1$ , $b = 5$ , and $c = - 47$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 1 \cdot - 47)}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot - 47}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 47$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot - 47}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 47$ .

$x = \frac{- 5 \pm \sqrt{25 - 188}}{2 \cdot - 1}$

Step 5.1.3

Subtract $188$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 163}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $- 163$ as $- 1 (163)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 163}}{2 \cdot - 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (163)}$ as $\sqrt{- 1} \cdot \sqrt{163}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{163}}{2 \cdot - 1}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 5 \pm i \sqrt{163}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm i \sqrt{163}}{- 2}$

Step 5.3

Simplify $\frac{- 5 \pm i \sqrt{163}}{- 2}$ .

$x = \frac{5 \pm i \sqrt{163}}{2}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot - 47}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot - 47$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot - 47}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $- 47$ .

$x = \frac{- 5 \pm \sqrt{25 - 188}}{2 \cdot - 1}$

Step 6.1.3

Subtract $188$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 163}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $- 163$ as $- 1 (163)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 163}}{2 \cdot - 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (163)}$ as $\sqrt{- 1} \cdot \sqrt{163}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{163}}{2 \cdot - 1}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 5 \pm i \sqrt{163}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm i \sqrt{163}}{- 2}$

Step 6.3

Simplify $\frac{- 5 \pm i \sqrt{163}}{- 2}$ .

$x = \frac{5 \pm i \sqrt{163}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{163}}{2}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot - 47}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 47$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot - 47}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 47$ .

$x = \frac{- 5 \pm \sqrt{25 - 188}}{2 \cdot - 1}$

Step 7.1.3

Subtract $188$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 163}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $- 163$ as $- 1 (163)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 163}}{2 \cdot - 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (163)}$ as $\sqrt{- 1} \cdot \sqrt{163}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{163}}{2 \cdot - 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 5 \pm i \sqrt{163}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm i \sqrt{163}}{- 2}$

Step 7.3

Simplify $\frac{- 5 \pm i \sqrt{163}}{- 2}$ .

$x = \frac{5 \pm i \sqrt{163}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{163}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{163}}{2}, \frac{5 - i \sqrt{163}}{2}$