Algebra Examples

$f (x) = \frac{1}{2} (x - 3) (x - 7)$

Step 1

Set $\frac{1}{2} \cdot ((x - 3) (x - 7))$ equal to $0$ .

$\frac{1}{2} \cdot ((x - 3) (x - 7)) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot ((x - 3) (x - 7))) = 2 \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $2 (\frac{1}{2} \cdot ((x - 3) (x - 7)))$ .

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Step 2.2.1.1.1

Expand $(x - 3) (x - 7)$ using the FOIL Method.

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Step 2.2.1.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x (x - 7) - 3 (x - 7))) = 2 \cdot 0$

Step 2.2.1.1.1.2

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x \cdot x + x \cdot - 7 - 3 (x - 7))) = 2 \cdot 0$

Step 2.2.1.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x \cdot x + x \cdot - 7 - 3 x - 3 \cdot - 7)) = 2 \cdot 0$

Step 2.2.1.1.2

Simplify and combine like terms.

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Step 2.2.1.1.2.1

Simplify each term.

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Step 2.2.1.1.2.1.1

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot (x^{2} + x \cdot - 7 - 3 x - 3 \cdot - 7)) = 2 \cdot 0$

Step 2.2.1.1.2.1.2

Move $- 7$ to the left of $x$ .

$2 (\frac{1}{2} \cdot (x^{2} - 7 \cdot x - 3 x - 3 \cdot - 7)) = 2 \cdot 0$

Step 2.2.1.1.2.1.3

Multiply $- 3$ by $- 7$ .

$2 (\frac{1}{2} \cdot (x^{2} - 7 x - 3 x + 21)) = 2 \cdot 0$

Step 2.2.1.1.2.2

Subtract $3 x$ from $- 7 x$ .

$2 (\frac{1}{2} \cdot (x^{2} - 10 x + 21)) = 2 \cdot 0$

Step 2.2.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} x^{2} + \frac{1}{2} (- 10 x) + \frac{1}{2} \cdot 21) = 2 \cdot 0$

Step 2.2.1.1.4

Simplify.

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Step 2.2.1.1.4.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{2}}{2} + \frac{1}{2} (- 10 x) + \frac{1}{2} \cdot 21) = 2 \cdot 0$

Step 2.2.1.1.4.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.2.1

Factor $2$ out of $- 10 x$ .

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (- 5 x)) + \frac{1}{2} \cdot 21) = 2 \cdot 0$

Step 2.2.1.1.4.2.2

Cancel the common factor.

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (- 5 x)) + \frac{1}{2} \cdot 21) = 2 \cdot 0$

Step 2.2.1.1.4.2.3

Rewrite the expression.

$2 (\frac{x^{2}}{2} - 5 x + \frac{1}{2} \cdot 21) = 2 \cdot 0$

Step 2.2.1.1.4.3

Combine $\frac{1}{2}$ and $21$ .

$2 (\frac{x^{2}}{2} - 5 x + \frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.5

Apply the distributive property.

$2 \frac{x^{2}}{2} + 2 (- 5 x) + 2 (\frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.1.1

Cancel the common factor.

$2 \frac{x^{2}}{2} + 2 (- 5 x) + 2 (\frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.6.1.2

Rewrite the expression.

$x^{2} + 2 (- 5 x) + 2 (\frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.6.2

Multiply $- 5$ by $2$ .

$x^{2} - 10 x + 2 (\frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.6.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.3.1

Cancel the common factor.

$x^{2} - 10 x + 2 (\frac{21}{2}) = 2 \cdot 0$

Step 2.2.1.1.6.3.2

Rewrite the expression.

$x^{2} - 10 x + 21 = 2 \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $2$ by $0$ .

$x^{2} - 10 x + 21 = 0$

Step 2.3

Factor $x^{2} - 10 x + 21$ using the AC method.

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Step 2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 2.3.2

Write the factored form using these integers.

$(x - 7) (x - 3) = 0$

Step 2.4

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

$x - 7 = 0$

$x - 3 = 0$

Step 2.5

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 2.5.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.6

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

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Step 2.6.1

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

$x - 3 = 0$

Step 2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.7

The final solution is all the values that make $(x - 7) (x - 3) = 0$ true.

$x = 7, 3$

Step 3