Basic Math Examples

$8 q (q - 10) + 7 = - 42 - q (q + 38)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$8 q \cdot q + 8 q \cdot - 10 + 7 = - 42 - q (q + 38)$

Step 1.2

Multiply $q$ by $q$ by adding the exponents.

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Step 1.2.1

Move $q$ .

$8 (q \cdot q) + 8 q \cdot - 10 + 7 = - 42 - q (q + 38)$

Step 1.2.2

Multiply $q$ by $q$ .

$8 q^{2} + 8 q \cdot - 10 + 7 = - 42 - q (q + 38)$

Step 1.3

Multiply $- 10$ by $8$ .

$8 q^{2} - 80 q + 7 = - 42 - q (q + 38)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$8 q^{2} - 80 q + 7 = - 42 - q \cdot q - q \cdot 38$

Step 2.2

Multiply $q$ by $q$ by adding the exponents.

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Step 2.2.1

Move $q$ .

$8 q^{2} - 80 q + 7 = - 42 - (q \cdot q) - q \cdot 38$

Step 2.2.2

Multiply $q$ by $q$ .

$8 q^{2} - 80 q + 7 = - 42 - q^{2} - q \cdot 38$

Step 2.3

Multiply $38$ by $- 1$ .

$8 q^{2} - 80 q + 7 = - 42 - q^{2} - 38 q$

Step 3

Move all terms containing $q$ to the left side of the equation.

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Step 3.1

Add $q^{2}$ to both sides of the equation.

$8 q^{2} - 80 q + 7 + q^{2} = - 42 - 38 q$

Step 3.2

Add $38 q$ to both sides of the equation.

$8 q^{2} - 80 q + 7 + q^{2} + 38 q = - 42$

Step 3.3

Add $8 q^{2}$ and $q^{2}$ .

$9 q^{2} - 80 q + 7 + 38 q = - 42$

Step 3.4

Add $- 80 q$ and $38 q$ .

$9 q^{2} - 42 q + 7 = - 42$

Step 4

Add $42$ to both sides of the equation.

$9 q^{2} - 42 q + 7 + 42 = 0$

Step 5

Add $7$ and $42$ .

$9 q^{2} - 42 q + 49 = 0$

Step 6

Factor using the perfect square rule.

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Step 6.1

Rewrite $9 q^{2}$ as ${(3 q)}^{2}$ .

${(3 q)}^{2} - 42 q + 49 = 0$

Step 6.2

Rewrite $49$ as $7^{2}$ .

${(3 q)}^{2} - 42 q + 7^{2} = 0$

Step 6.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$42 q = 2 \cdot (3 q) \cdot 7$

Step 6.4

Rewrite the polynomial.

${(3 q)}^{2} - 2 \cdot (3 q) \cdot 7 + 7^{2} = 0$

Step 6.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 q$ and $b = 7$ .

${(3 q - 7)}^{2} = 0$

Step 7

Set the $3 q - 7$ equal to $0$ .

$3 q - 7 = 0$

Step 8

Solve for $q$ .

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Step 8.1

Add $7$ to both sides of the equation.

$3 q = 7$

Step 8.2

Divide each term in $3 q = 7$ by $3$ and simplify.

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Step 8.2.1

Divide each term in $3 q = 7$ by $3$ .

$\frac{3 q}{3} = \frac{7}{3}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{3 q}{3} = \frac{7}{3}$

Step 8.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{7}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$q = \frac{7}{3}$

Decimal Form:

$q = 2.\overline{3}$

Mixed Number Form:

$q = 2 \frac{1}{3}$