Precalculus Examples

$3 x^{2} + 6 x - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$3 x^{2} + 6 x = 5$

Step 2

Divide each term in $3 x^{2} + 6 x = 5$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} + 6 x = 5$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{6 x}{3} = \frac{5}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{6 x}{3} = \frac{5}{3}$

Step 2.2.1.1.2

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

$x^{2} + \frac{6 x}{3} = \frac{5}{3}$

Step 2.2.1.2

Cancel the common factor of $6$ and $3$ .

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Step 2.2.1.2.1

Factor $3$ out of $6 x$ .

$x^{2} + \frac{3 (2 x)}{3} = \frac{5}{3}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $3$ out of $3$ .

$x^{2} + \frac{3 (2 x)}{3 (1)} = \frac{5}{3}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{3 (2 x)}{3 \cdot 1} = \frac{5}{3}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{2 x}{1} = \frac{5}{3}$

Step 2.2.1.2.2.4

Divide $2 x$ by $1$ .

$x^{2} + 2 x = \frac{5}{3}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(1)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + 2 x + {(1)}^{2} = \frac{5}{3} + {(1)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

One to any power is one.

$x^{2} + 2 x + 1 = \frac{5}{3} + {(1)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{5}{3} + {(1)}^{2}$ .

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Step 5.2.1.1

One to any power is one.

$x^{2} + 2 x + 1 = \frac{5}{3} + 1$

Step 5.2.1.2

Write $1$ as a fraction with a common denominator.

$x^{2} + 2 x + 1 = \frac{5}{3} + \frac{3}{3}$

Step 5.2.1.3

Combine the numerators over the common denominator.

$x^{2} + 2 x + 1 = \frac{5 + 3}{3}$

Step 5.2.1.4

Add $5$ and $3$ .

$x^{2} + 2 x + 1 = \frac{8}{3}$

Step 6

Factor the perfect trinomial square into ${(x + 1)}^{2}$ .

${(x + 1)}^{2} = \frac{8}{3}$

Step 7

Solve the equation for $x$ .

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Step 7.1

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

$x + 1 = \pm \sqrt{\frac{8}{3}}$

Step 7.2

Simplify $\pm \sqrt{\frac{8}{3}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{8}{3}}$ as $\frac{\sqrt{8}}{\sqrt{3}}$ .

$x + 1 = \pm \frac{\sqrt{8}}{\sqrt{3}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.2.2.1.1

Factor $4$ out of $8$ .

$x + 1 = \pm \frac{\sqrt{4 (2)}}{\sqrt{3}}$

Step 7.2.2.1.2

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

$x + 1 = \pm \frac{\sqrt{2^{2} \cdot 2}}{\sqrt{3}}$

Step 7.2.2.2

Pull terms out from under the radical.

$x + 1 = \pm \frac{2 \sqrt{2}}{\sqrt{3}}$

Step 7.2.3

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x + 1 = \pm \frac{2 \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.2.4

Combine and simplify the denominator.

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Step 7.2.4.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.2.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.2.4.5

Add $1$ and $1$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.2.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.2.4.6.4

Cancel the common factor of $2$ .

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Step 7.2.4.6.4.1

Cancel the common factor.

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.2.4.6.4.2

Rewrite the expression.

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3^{1}}$

Step 7.2.4.6.5

Evaluate the exponent.

$x + 1 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Combine using the product rule for radicals.

$x + 1 = \pm \frac{2 \sqrt{3 \cdot 2}}{3}$

Step 7.2.5.2

Multiply $3$ by $2$ .

$x + 1 = \pm \frac{2 \sqrt{6}}{3}$

Step 7.3

Subtract $1$ from both sides of the equation.

$x = \pm \frac{2 \sqrt{6}}{3} - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{2 \sqrt{6}}{3} - 1$

Decimal Form:

$x = 0.63299316 \dots, - 2.63299316 \dots$