Pre-Algebra Examples

$a (9 a + 2) = 7$

Step 1

Simplify $a (9 a + 2)$ .

Tap for more steps...

Step 1.1

Simplify by multiplying through.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$a (9 a) + a \cdot 2 = 7$

Step 1.1.2

Reorder.

Tap for more steps...

Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$9 a \cdot a + a \cdot 2 = 7$

Step 1.1.2.2

Move $2$ to the left of $a$ .

$9 a \cdot a + 2 \cdot a = 7$

Step 1.2

Multiply $a$ by $a$ by adding the exponents.

Tap for more steps...

Step 1.2.1

Move $a$ .

$9 (a \cdot a) + 2 \cdot a = 7$

Step 1.2.2

Multiply $a$ by $a$ .

$9 a^{2} + 2 \cdot a = 7$

$9 a^{2} + 2 a = 7$

Step 2

Subtract $7$ from both sides of the equation.

$9 a^{2} + 2 a - 7 = 0$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

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 $a \cdot c = 9 \cdot - 7 = - 63$ and whose sum is $b = 2$ .

Tap for more steps...

Step 3.1.1

Factor $2$ out of $2 a$ .

$9 a^{2} + 2 (a) - 7 = 0$

Step 3.1.2

Rewrite $2$ as $- 7$ plus $9$

$9 a^{2} + (- 7 + 9) a - 7 = 0$

Step 3.1.3

Apply the distributive property.

$9 a^{2} - 7 a + 9 a - 7 = 0$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$(9 a^{2} - 7 a) + 9 a - 7 = 0$

Step 3.2.2

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

$a (9 a - 7) + 1 (9 a - 7) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $9 a - 7$ .

$(9 a - 7) (a + 1) = 0$

Step 4

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

$9 a - 7 = 0$

$a + 1 = 0$

Step 5

Set $9 a - 7$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 5.1

Set $9 a - 7$ equal to $0$ .

$9 a - 7 = 0$

Step 5.2

Solve $9 a - 7 = 0$ for $a$ .

Tap for more steps...

Step 5.2.1

Add $7$ to both sides of the equation.

$9 a = 7$

Step 5.2.2

Divide each term in $9 a = 7$ by $9$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in $9 a = 7$ by $9$ .

$\frac{9 a}{9} = \frac{7}{9}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{9 a}{9} = \frac{7}{9}$

Step 5.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{7}{9}$

Step 6

Set $a + 1$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 6.1

Set $a + 1$ equal to $0$ .

$a + 1 = 0$

Step 6.2

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 7

The final solution is all the values that make $(9 a - 7) (a + 1) = 0$ true.

$a = \frac{7}{9}, - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = \frac{7}{9}, - 1$

Decimal Form:

$a = 0.\overline{7}, - 1$