Precalculus Examples

${(x - 3)}^{2} = 3$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $3$ from both sides of the equation.

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

Step 1.2

Simplify ${(x - 3)}^{2} - 3$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$(x - 3) (x - 3) - 3 = 0$

Step 1.2.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$x (x - 3) - 3 (x - 3) - 3 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) - 3 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 = 0$

Step 1.2.1.3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 3 = 0$

Step 1.2.1.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 - 3 = 0$

Step 1.2.1.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 - 3 = 0$

Step 1.2.2

Subtract $3$ from $9$ .

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

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

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

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 $2$ .

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

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 1}$

Step 4.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.1.4.2

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

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

Step 4.1.5

Pull terms out from under the radical.

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

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{2}$ .

$x = 3 \pm \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3 \pm \sqrt{3}$

Decimal Form:

$x = 4.73205080 \dots, 1.26794919 \dots$