Algebra Examples

$x^{2} + \frac{3}{2} x - 3 = 0$

Step 1

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

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Step 1.1

Combine $\frac{3}{2}$ and $x$ .

$x^{2} + \frac{3 x}{2} - 3 = 0$

Step 1.2

Add $3$ to both sides of the equation.

$x^{2} + \frac{3 x}{2} = 3$

Step 2

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

${(\frac{b}{2})}^{2} = {(\frac{3}{4})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{3 x}{2} + {(\frac{3}{4})}^{2} = 3 + {(\frac{3}{4})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{3}{4}$ .

$x^{2} + \frac{3 x}{2} + \frac{3^{2}}{4^{2}} = 3 + {(\frac{3}{4})}^{2}$

Step 4.1.1.2

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

$x^{2} + \frac{3 x}{2} + \frac{9}{4^{2}} = 3 + {(\frac{3}{4})}^{2}$

Step 4.1.1.3

Raise $4$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 3 + {(\frac{3}{4})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $3 + {(\frac{3}{4})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{3}{4}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 3 + \frac{3^{2}}{4^{2}}$

Step 4.2.1.1.2

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

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 3 + \frac{9}{4^{2}}$

Step 4.2.1.1.3

Raise $4$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 3 + \frac{9}{16}$

Step 4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 3 \cdot \frac{16}{16} + \frac{9}{16}$

Step 4.2.1.3

Combine $3$ and $\frac{16}{16}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{3 \cdot 16}{16} + \frac{9}{16}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{3 \cdot 16 + 9}{16}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $3$ by $16$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{48 + 9}{16}$

Step 4.2.1.5.2

Add $48$ and $9$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{57}{16}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{3}{4})}^{2}$ .

${(x + \frac{3}{4})}^{2} = \frac{57}{16}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x + \frac{3}{4} = \pm \sqrt{\frac{57}{16}}$

Step 6.2

Simplify $\pm \sqrt{\frac{57}{16}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{57}{16}}$ as $\frac{\sqrt{57}}{\sqrt{16}}$ .

$x + \frac{3}{4} = \pm \frac{\sqrt{57}}{\sqrt{16}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$x + \frac{3}{4} = \pm \frac{\sqrt{57}}{\sqrt{4^{2}}}$

Step 6.2.2.2

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

$x + \frac{3}{4} = \pm \frac{\sqrt{57}}{4}$

Step 6.3

Subtract $\frac{3}{4}$ from both sides of the equation.

$x = \pm \frac{\sqrt{57}}{4} - \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{57}}{4} - \frac{3}{4}$

Decimal Form:

$x = 1.13745860 \dots, - 2.63745860 \dots$