Algebra Examples

$M (x) = (2 x - 3) (x^{2} + 3 x + 10)$

Step 1

Set $(2 x - 3) (x^{2} + 3 x + 10)$ equal to $0$ .

$(2 x - 3) (x^{2} + 3 x + 10) = 0$

Step 2

Solve for $x$ .

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Step 2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 3 = 0$

$x^{2} + 3 x + 10 = 0$

Step 2.2

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 2.2.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 2.2.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.2.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.2.2.2.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 2.2.2.2.2

Simplify the left side.

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Step 2.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3

Set $x^{2} + 3 x + 10$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 3 x + 10$ equal to $0$ .

$x^{2} + 3 x + 10 = 0$

Step 2.3.2

Solve $x^{2} + 3 x + 10 = 0$ for $x$ .

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Step 2.3.2.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 2.3.2.3

Simplify.

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Step 2.3.2.3.1

Simplify the numerator.

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Step 2.3.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 2.3.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 2.3.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 10}}{2 \cdot 1}$

Step 2.3.2.3.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{- 3 \pm \sqrt{9 - 40}}{2 \cdot 1}$

Step 2.3.2.3.1.3

Subtract $40$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 31}}{2 \cdot 1}$

Step 2.3.2.3.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 1}$

Step 2.3.2.3.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 1}$

Step 2.3.2.3.1.6

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

$x = \frac{- 3 \pm i \sqrt{31}}{2 \cdot 1}$

Step 2.3.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{31}}{2}$

Step 2.3.2.4

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{31}}{2}, - \frac{3 + i \sqrt{31}}{2}$

Step 2.4

The final solution is all the values that make $(2 x - 3) (x^{2} + 3 x + 10) = 0$ true.

$x = \frac{3}{2}, - \frac{3 - i \sqrt{31}}{2}, - \frac{3 + i \sqrt{31}}{2}$

Step 3