Finite Math Examples

$(4 x + 1) (x + 2) = - 2$

Step 1

Simplify the equation into a proper form to complete the square.

Tap for more steps...

Step 1.1

Simplify $(4 x + 1) (x + 2)$ .

Tap for more steps...

Step 1.1.1

Expand $(4 x + 1) (x + 2)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.1

Apply the distributive property.

$4 x (x + 2) + 1 (x + 2) = - 2$

Step 1.1.1.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 2 + 1 (x + 2) = - 2$

Step 1.1.1.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 2 + 1 x + 1 \cdot 2 = - 2$

Step 1.1.2

Simplify and combine like terms.

Tap for more steps...

Step 1.1.2.1

Simplify each term.

Tap for more steps...

Step 1.1.2.1.1

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

Tap for more steps...

Step 1.1.2.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot 2 + 1 x + 1 \cdot 2 = - 2$

Step 1.1.2.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.2.1.2

Multiply $2$ by $4$ .

$4 x^{2} + 8 x + 1 x + 1 \cdot 2 = - 2$

Step 1.1.2.1.3

Multiply $x$ by $1$ .

$4 x^{2} + 8 x + x + 1 \cdot 2 = - 2$

Step 1.1.2.1.4

Multiply $2$ by $1$ .

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

Step 1.1.2.2

Add $8 x$ and $x$ .

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

Step 1.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.2.1

Subtract $2$ from both sides of the equation.

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

Step 1.2.2

Subtract $2$ from $- 2$ .

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

Step 2

Divide each term in $4 x^{2} + 9 x = - 4$ by $4$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $4 x^{2} + 9 x = - 4$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{9 x}{4} = \frac{- 4}{4}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{9 x}{4} = \frac{- 4}{4}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{9 x}{4} = \frac{- 4}{4}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $- 4$ by $4$ .

$x^{2} + \frac{9 x}{4} = - 1$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{9}{8})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{9 x}{4} + {(\frac{9}{8})}^{2} = - 1 + {(\frac{9}{8})}^{2}$

Step 5

Simplify the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Simplify each term.

Tap for more steps...

Step 5.1.1.1

Apply the product rule to $\frac{9}{8}$ .

$x^{2} + \frac{9 x}{4} + \frac{9^{2}}{8^{2}} = - 1 + {(\frac{9}{8})}^{2}$

Step 5.1.1.2

Raise $9$ to the power of $2$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{8^{2}} = - 1 + {(\frac{9}{8})}^{2}$

Step 5.1.1.3

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

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = - 1 + {(\frac{9}{8})}^{2}$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $- 1 + {(\frac{9}{8})}^{2}$ .

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

Apply the product rule to $\frac{9}{8}$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = - 1 + \frac{9^{2}}{8^{2}}$

Step 5.2.1.1.2

Raise $9$ to the power of $2$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = - 1 + \frac{81}{8^{2}}$

Step 5.2.1.1.3

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

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = - 1 + \frac{81}{64}$

Step 5.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = - 1 \cdot \frac{64}{64} + \frac{81}{64}$

Step 5.2.1.3

Combine $- 1$ and $\frac{64}{64}$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = \frac{- 1 \cdot 64}{64} + \frac{81}{64}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = \frac{- 1 \cdot 64 + 81}{64}$

Step 5.2.1.5

Simplify the numerator.

Tap for more steps...

Step 5.2.1.5.1

Multiply $- 1$ by $64$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = \frac{- 64 + 81}{64}$

Step 5.2.1.5.2

Add $- 64$ and $81$ .

$x^{2} + \frac{9 x}{4} + \frac{81}{64} = \frac{17}{64}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{9}{8})}^{2}$ .

${(x + \frac{9}{8})}^{2} = \frac{17}{64}$

Step 7

Solve the equation for $x$ .

Tap for more steps...

Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{9}{8} = \pm \sqrt{\frac{17}{64}}$

Step 7.2

Simplify $\pm \sqrt{\frac{17}{64}}$ .

Tap for more steps...

Step 7.2.1

Rewrite $\sqrt{\frac{17}{64}}$ as $\frac{\sqrt{17}}{\sqrt{64}}$ .

$x + \frac{9}{8} = \pm \frac{\sqrt{17}}{\sqrt{64}}$

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

Rewrite $64$ as $8^{2}$ .

$x + \frac{9}{8} = \pm \frac{\sqrt{17}}{\sqrt{8^{2}}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{9}{8} = \pm \frac{\sqrt{17}}{8}$

Step 7.3

Subtract $\frac{9}{8}$ from both sides of the equation.

$x = \pm \frac{\sqrt{17}}{8} - \frac{9}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{17}}{8} - \frac{9}{8}$

Decimal Form:

$x = - 0.60961179 \dots, - 1.64038820 \dots$