Algebra Examples

$f (h) = h (h - 2) (2 h - 1)$

Step 1

Set $h (h - 2) (2 h - 1)$ equal to $0$ .

$h (h - 2) (2 h - 1) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $h (h - 2) (2 h - 1)$ .

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Step 2.1.1

Simplify by multiplying through.

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Step 2.1.1.1

Apply the distributive property.

$(h \cdot h + h \cdot - 2) (2 h - 1) = 0$

Step 2.1.1.2

Simplify the expression.

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Step 2.1.1.2.1

Multiply $h$ by $h$ .

$(h^{2} + h \cdot - 2) (2 h - 1) = 0$

Step 2.1.1.2.2

Move $- 2$ to the left of $h$ .

$(h^{2} - 2 \cdot h) (2 h - 1) = 0$

Step 2.1.2

Expand $(h^{2} - 2 h) (2 h - 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$h^{2} (2 h - 1) - 2 h (2 h - 1) = 0$

Step 2.1.2.2

Apply the distributive property.

$h^{2} (2 h) + h^{2} \cdot - 1 - 2 h (2 h - 1) = 0$

Step 2.1.2.3

Apply the distributive property.

$h^{2} (2 h) + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 h^{2} h + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.2

Multiply $h^{2}$ by $h$ by adding the exponents.

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Step 2.1.3.1.2.1

Move $h$ .

$2 (h \cdot h^{2}) + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.2.2

Multiply $h$ by $h^{2}$ .

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Step 2.1.3.1.2.2.1

Raise $h$ to the power of $1$ .

$2 (h^{1} h^{2}) + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 h^{1 + 2} + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.2.3

Add $1$ and $2$ .

$2 h^{3} + h^{2} \cdot - 1 - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.3

Move $- 1$ to the left of $h^{2}$ .

$2 h^{3} - 1 \cdot h^{2} - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.4

Rewrite $- 1 h^{2}$ as $- h^{2}$ .

$2 h^{3} - h^{2} - 2 h (2 h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.5

Rewrite using the commutative property of multiplication.

$2 h^{3} - h^{2} - 2 \cdot 2 h \cdot h - 2 h \cdot - 1 = 0$

Step 2.1.3.1.6

Multiply $h$ by $h$ by adding the exponents.

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Step 2.1.3.1.6.1

Move $h$ .

$2 h^{3} - h^{2} - 2 \cdot 2 (h \cdot h) - 2 h \cdot - 1 = 0$

Step 2.1.3.1.6.2

Multiply $h$ by $h$ .

$2 h^{3} - h^{2} - 2 \cdot 2 h^{2} - 2 h \cdot - 1 = 0$

Step 2.1.3.1.7

Multiply $- 2$ by $2$ .

$2 h^{3} - h^{2} - 4 h^{2} - 2 h \cdot - 1 = 0$

Step 2.1.3.1.8

Multiply $- 1$ by $- 2$ .

$2 h^{3} - h^{2} - 4 h^{2} + 2 h = 0$

Step 2.1.3.2

Subtract $4 h^{2}$ from $- h^{2}$ .

$2 h^{3} - 5 h^{2} + 2 h = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $h$ out of $2 h^{3} - 5 h^{2} + 2 h$ .

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Step 2.2.1.1

Factor $h$ out of $2 h^{3}$ .

$h (2 h^{2}) - 5 h^{2} + 2 h = 0$

Step 2.2.1.2

Factor $h$ out of $- 5 h^{2}$ .

$h (2 h^{2}) + h (- 5 h) + 2 h = 0$

Step 2.2.1.3

Factor $h$ out of $2 h$ .

$h (2 h^{2}) + h (- 5 h) + h \cdot 2 = 0$

Step 2.2.1.4

Factor $h$ out of $h (2 h^{2}) + h (- 5 h)$ .

$h (2 h^{2} - 5 h) + h \cdot 2 = 0$

Step 2.2.1.5

Factor $h$ out of $h (2 h^{2} - 5 h) + h \cdot 2$ .

$h (2 h^{2} - 5 h + 2) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor by grouping.

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Step 2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 2.2.2.1.1.1

Factor $- 5$ out of $- 5 h$ .

$h (2 h^{2} - 5 h + 2) = 0$

Step 2.2.2.1.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

$h (2 h^{2} + (- 1 - 4) h + 2) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$h (2 h^{2} - 1 h - 4 h + 2) = 0$

Step 2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1.2.1

Group the first two terms and the last two terms.

$h ((2 h^{2} - 1 h) - 4 h + 2) = 0$

Step 2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$h (h (2 h - 1) - 2 (2 h - 1)) = 0$

Step 2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 h - 1$ .

$h ((2 h - 1) (h - 2)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$h (2 h - 1) (h - 2) = 0$

Step 2.3

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

$h = 0$

$2 h - 1 = 0$

$h - 2 = 0$

Step 2.4

Set $h$ equal to $0$ .

$h = 0$

Step 2.5

Set $2 h - 1$ equal to $0$ and solve for $h$ .

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Step 2.5.1

Set $2 h - 1$ equal to $0$ .

$2 h - 1 = 0$

Step 2.5.2

Solve $2 h - 1 = 0$ for $h$ .

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Step 2.5.2.1

Add $1$ to both sides of the equation.

$2 h = 1$

Step 2.5.2.2

Divide each term in $2 h = 1$ by $2$ and simplify.

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Step 2.5.2.2.1

Divide each term in $2 h = 1$ by $2$ .

$\frac{2 h}{2} = \frac{1}{2}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 h}{2} = \frac{1}{2}$

Step 2.5.2.2.2.1.2

Divide $h$ by $1$ .

$h = \frac{1}{2}$

Step 2.6

Set $h - 2$ equal to $0$ and solve for $h$ .

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Step 2.6.1

Set $h - 2$ equal to $0$ .

$h - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$h = 2$

Step 2.7

The final solution is all the values that make $h (2 h - 1) (h - 2) = 0$ true.

$h = 0, \frac{1}{2}, 2$

Step 3