Algebra Examples

$(x - 3) (x^{2} - 22 x + 84) = 0$

Step 1

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

$x - 3 = 0$

$x^{2} - 22 x + 84 = 0$

Step 2

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

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Step 2.1

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

$x - 3 = 0$

Step 2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3

Set $x^{2} - 22 x + 84$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2} - 22 x + 84$ equal to $0$ .

$x^{2} - 22 x + 84 = 0$

Step 3.2

Solve $x^{2} - 22 x + 84 = 0$ for $x$ .

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Step 3.2.1

Use the quadratic formula to find the solutions.

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

Step 3.2.2

Substitute the values $a = 1$ , $b = - 22$ , and $c = 84$ into the quadratic formula and solve for $x$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (1 \cdot 84)}}{2 \cdot 1}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Simplify the numerator.

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Step 3.2.3.1.1

Raise $- 22$ to the power of $2$ .

$x = \frac{22 \pm \sqrt{484 - 4 \cdot 1 \cdot 84}}{2 \cdot 1}$

Step 3.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 84$ .

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Step 3.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{22 \pm \sqrt{484 - 4 \cdot 84}}{2 \cdot 1}$

Step 3.2.3.1.2.2

Multiply $- 4$ by $84$ .

$x = \frac{22 \pm \sqrt{484 - 336}}{2 \cdot 1}$

Step 3.2.3.1.3

Subtract $336$ from $484$ .

$x = \frac{22 \pm \sqrt{148}}{2 \cdot 1}$

Step 3.2.3.1.4

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 3.2.3.1.4.1

Factor $4$ out of $148$ .

$x = \frac{22 \pm \sqrt{4 (37)}}{2 \cdot 1}$

Step 3.2.3.1.4.2

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

$x = \frac{22 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 1}$

Step 3.2.3.1.5

Pull terms out from under the radical.

$x = \frac{22 \pm 2 \sqrt{37}}{2 \cdot 1}$

Step 3.2.3.2

Multiply $2$ by $1$ .

$x = \frac{22 \pm 2 \sqrt{37}}{2}$

Step 3.2.3.3

Simplify $\frac{22 \pm 2 \sqrt{37}}{2}$ .

$x = 11 \pm \sqrt{37}$

Step 3.2.4

The final answer is the combination of both solutions.

$x = 11 + \sqrt{37}, 11 - \sqrt{37}$

Step 4

The final solution is all the values that make $(x - 3) (x^{2} - 22 x + 84) = 0$ true.

$x = 3, 11 + \sqrt{37}, 11 - \sqrt{37}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3, 11 + \sqrt{37}, 11 - \sqrt{37}$

Decimal Form:

$x = 3, 17.08276253 \dots, 4.91723746 \dots$