Algebra Examples

$\frac{17}{10} x^{2} + 3 x + 5$

Step 1

Set $\frac{17}{10} x^{2} + 3 x + 5$ equal to $0$ .

$\frac{17}{10} x^{2} + 3 x + 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Combine $\frac{17}{10}$ and $x^{2}$ .

$\frac{17 x^{2}}{10} + 3 x + 5 = 0$

Step 2.2

Multiply through by the least common denominator $10$ , then simplify.

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Step 2.2.1

Apply the distributive property.

$10 (\frac{17 x^{2}}{10}) + 10 (3 x) + 10 \cdot 5 = 0$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $10$ .

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Step 2.2.2.1.1

Cancel the common factor.

$10 (\frac{17 x^{2}}{10}) + 10 (3 x) + 10 \cdot 5 = 0$

Step 2.2.2.1.2

Rewrite the expression.

$17 x^{2} + 10 (3 x) + 10 \cdot 5 = 0$

Step 2.2.2.2

Multiply $3$ by $10$ .

$17 x^{2} + 30 x + 10 \cdot 5 = 0$

Step 2.2.2.3

Multiply $10$ by $5$ .

$17 x^{2} + 30 x + 50 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 17$ , $b = 30$ , and $c = 50$ into the quadratic formula and solve for $x$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (17 \cdot 50)}}{2 \cdot 17}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 17 \cdot 50}}{2 \cdot 17}$

Step 2.5.1.2

Multiply $- 4 \cdot 17 \cdot 50$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $17$ .

$x = \frac{- 30 \pm \sqrt{900 - 68 \cdot 50}}{2 \cdot 17}$

Step 2.5.1.2.2

Multiply $- 68$ by $50$ .

$x = \frac{- 30 \pm \sqrt{900 - 3400}}{2 \cdot 17}$

Step 2.5.1.3

Subtract $3400$ from $900$ .

$x = \frac{- 30 \pm \sqrt{- 2500}}{2 \cdot 17}$

Step 2.5.1.4

Rewrite $- 2500$ as $- 1 (2500)$ .

$x = \frac{- 30 \pm \sqrt{- 1 \cdot 2500}}{2 \cdot 17}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (2500)}$ as $\sqrt{- 1} \cdot \sqrt{2500}$ .

$x = \frac{- 30 \pm \sqrt{- 1} \cdot \sqrt{2500}}{2 \cdot 17}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 30 \pm i \cdot \sqrt{2500}}{2 \cdot 17}$

Step 2.5.1.7

Rewrite $2500$ as $50^{2}$ .

$x = \frac{- 30 \pm i \cdot \sqrt{50^{2}}}{2 \cdot 17}$

Step 2.5.1.8

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

$x = \frac{- 30 \pm i \cdot 50}{2 \cdot 17}$

Step 2.5.1.9

Move $50$ to the left of $i$ .

$x = \frac{- 30 \pm 50 i}{2 \cdot 17}$

Step 2.5.2

Multiply $2$ by $17$ .

$x = \frac{- 30 \pm 50 i}{34}$

Step 2.5.3

Simplify $\frac{- 30 \pm 50 i}{34}$ .

$x = \frac{- 15 \pm 25 i}{17}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{15}{17} + \frac{25 i}{17}, - \frac{15}{17} - \frac{25 i}{17}$

$x = \frac{- 15 \pm 25 i}{17}$

Step 3