Pre-Algebra Examples

$420 = (2 x + 3) x$

Step 1

Rewrite the equation as $(2 x + 3) x = 420$ .

$(2 x + 3) x = 420$

Step 2

Simplify $(2 x + 3) x$ .

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Step 2.1

Apply the distributive property.

$2 x \cdot x + 3 x = 420$

Step 2.2

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

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Step 2.2.1

Move $x$ .

$2 (x \cdot x) + 3 x = 420$

Step 2.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 3 x = 420$

Step 3

Subtract $420$ from both sides of the equation.

$2 x^{2} + 3 x - 420 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 2$ , $b = 3$ , and $c = - 420$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 420)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 420}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 420$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 420}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 420$ .

$x = \frac{- 3 \pm \sqrt{9 + 3360}}{2 \cdot 2}$

Step 6.1.3

Add $9$ and $3360$ .

$x = \frac{- 3 \pm \sqrt{3369}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{3369}}{4}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 420}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 420$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 420}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 420$ .

$x = \frac{- 3 \pm \sqrt{9 + 3360}}{2 \cdot 2}$

Step 7.1.3

Add $9$ and $3360$ .

$x = \frac{- 3 \pm \sqrt{3369}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{3369}}{4}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{3369}}{4}$

Step 7.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{3369}}{4}$

Step 7.5

Factor $- 1$ out of $\sqrt{3369}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{3369})}{4}$

Step 7.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{3369})$ .

$x = \frac{- 1 (3 - \sqrt{3369})}{4}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{3369}}{4}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 420}}{2 \cdot 2}$

Step 8.1.2

Multiply $- 4 \cdot 2 \cdot - 420$ .

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Step 8.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 420}}{2 \cdot 2}$

Step 8.1.2.2

Multiply $- 8$ by $- 420$ .

$x = \frac{- 3 \pm \sqrt{9 + 3360}}{2 \cdot 2}$

Step 8.1.3

Add $9$ and $3360$ .

$x = \frac{- 3 \pm \sqrt{3369}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{3369}}{4}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{3369}}{4}$

Step 8.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{3369}}{4}$

Step 8.5

Factor $- 1$ out of $- \sqrt{3369}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{3369})}{4}$

Step 8.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{3369})$ .

$x = \frac{- 1 (3 + \sqrt{3369})}{4}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{3369}}{4}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{3369}}{4}, - \frac{3 + \sqrt{3369}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{3369}}{4}, - \frac{3 + \sqrt{3369}}{4}$

Decimal Form:

$x = 13.76077186 \dots, - 15.26077186 \dots$