Algebra Examples

$4.5 x^{2} + 3 x + \frac{1}{2} = 0$

Step 1

Multiply through by the least common denominator $2$ , then simplify.

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Step 1.1

Apply the distributive property.

$2 (4.5 x^{2}) + 2 (3 x) + 2 (\frac{1}{2}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $4.5$ by $2$ .

$9 x^{2} + 2 (3 x) + 2 (\frac{1}{2}) = 0$

Step 1.2.2

Multiply $3$ by $2$ .

$9 x^{2} + 6 x + 2 (\frac{1}{2}) = 0$

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Cancel the common factor.

$9 x^{2} + 6 x + 2 (\frac{1}{2}) = 0$

Step 1.2.3.2

Rewrite the expression.

$9 x^{2} + 6 x + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 9$ , $b = 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (9 \cdot 1)}}{2 \cdot 9}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 9 \cdot 1}}{2 \cdot 9}$

Step 4.1.2

Multiply $- 4 \cdot 9 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 6 \pm \sqrt{36 - 36 \cdot 1}}{2 \cdot 9}$

Step 4.1.2.2

Multiply $- 36$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 36}}{2 \cdot 9}$

Step 4.1.3

Subtract $36$ from $36$ .

$x = \frac{- 6 \pm \sqrt{0}}{2 \cdot 9}$

Step 4.1.4

Rewrite $0$ as $0^{2}$ .

$x = \frac{- 6 \pm \sqrt{0^{2}}}{2 \cdot 9}$

Step 4.1.5

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

$x = \frac{- 6 \pm 0}{2 \cdot 9}$

Step 4.2

Multiply $2$ by $9$ .

$x = \frac{- 6 \pm 0}{18}$

Step 4.3

Simplify $\frac{- 6 \pm 0}{18}$ .

$x = \frac{- 6}{18}$

Step 4.4

Cancel the common factor of $- 6$ and $18$ .

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Step 4.4.1

Factor $6$ out of $- 6$ .

$x = \frac{6 (- 1)}{18}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $6$ out of $18$ .

$x = \frac{6 \cdot - 1}{6 \cdot 3}$

Step 4.4.2.2

Cancel the common factor.

$x = \frac{6 \cdot - 1}{6 \cdot 3}$

Step 4.4.2.3

Rewrite the expression.

$x = \frac{- 1}{3}$

Step 4.5

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4.6

Divide $1$ by $3$ .

$x = - 1 \cdot 0.\overline{3}$

Step 4.7

Multiply $- 1$ by $0.\overline{3}$ .

$x = - 0.\overline{3}$

Step 5

The final answer is the combination of both solutions.

$x = - 0.\overline{3}$ Double roots