Basic Math Examples

$\frac{1}{20} a^{2} - \frac{3}{2} a + 10 = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{20}$ and $a^{2}$ .

$\frac{a^{2}}{20} - \frac{3}{2} a + 10 = 0$

Step 1.2

Combine $a$ and $\frac{3}{2}$ .

$\frac{a^{2}}{20} - \frac{a \cdot 3}{2} + 10 = 0$

Step 1.3

Move $3$ to the left of $a$ .

$\frac{a^{2}}{20} - \frac{3 a}{2} + 10 = 0$

Step 2

Multiply each term in $\frac{a^{2}}{20} - \frac{3 a}{2} + 10 = 0$ by $20$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{a^{2}}{20} - \frac{3 a}{2} + 10 = 0$ by $20$ .

$\frac{a^{2}}{20} \cdot 20 - \frac{3 a}{2} \cdot 20 + 10 \cdot 20 = 0 \cdot 20$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $20$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{a^{2}}{20} \cdot 20 - \frac{3 a}{2} \cdot 20 + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.1.2

Rewrite the expression.

$a^{2} - \frac{3 a}{2} \cdot 20 + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{3 a}{2}$ into the numerator.

$a^{2} + \frac{- 3 a}{2} \cdot 20 + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.2.2

Factor $2$ out of $20$ .

$a^{2} + \frac{- 3 a}{2} \cdot (2 (10)) + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.2.3

Cancel the common factor.

$a^{2} + \frac{- 3 a}{2} \cdot (2 \cdot 10) + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.2.4

Rewrite the expression.

$a^{2} - 3 a \cdot 10 + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.3

Multiply $10$ by $- 3$ .

$a^{2} - 30 a + 10 \cdot 20 = 0 \cdot 20$

Step 2.2.1.4

Multiply $10$ by $20$ .

$a^{2} - 30 a + 200 = 0 \cdot 20$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $20$ .

$a^{2} - 30 a + 200 = 0$

Step 3

Factor $a^{2} - 30 a + 200$ using the AC method.

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Step 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 $200$ and whose sum is $- 30$ .

$- 20, - 10$

Step 3.2

Write the factored form using these integers.

$(a - 20) (a - 10) = 0$

Step 4

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

$a - 20 = 0$

$a - 10 = 0$

Step 5

Set $a - 20$ equal to $0$ and solve for $a$ .

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Step 5.1

Set $a - 20$ equal to $0$ .

$a - 20 = 0$

Step 5.2

Add $20$ to both sides of the equation.

$a = 20$

Step 6

Set $a - 10$ equal to $0$ and solve for $a$ .

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Step 6.1

Set $a - 10$ equal to $0$ .

$a - 10 = 0$

Step 6.2

Add $10$ to both sides of the equation.

$a = 10$

Step 7

The final solution is all the values that make $(a - 20) (a - 10) = 0$ true.

$a = 20, 10$