Algebra Examples

${(3 x - 8)}^{2} = 3 x^{2} - 8 x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 x^{2} - 8 x = {(3 x - 8)}^{2}$

Step 2

Simplify ${(3 x - 8)}^{2}$ .

Tap for more steps...

Step 2.1

Rewrite ${(3 x - 8)}^{2}$ as $(3 x - 8) (3 x - 8)$ .

$3 x^{2} - 8 x = (3 x - 8) (3 x - 8)$

Step 2.2

Expand $(3 x - 8) (3 x - 8)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$3 x^{2} - 8 x = 3 x (3 x - 8) - 8 (3 x - 8)$

Step 2.2.2

Apply the distributive property.

$3 x^{2} - 8 x = 3 x (3 x) + 3 x \cdot - 8 - 8 (3 x - 8)$

Step 2.2.3

Apply the distributive property.

$3 x^{2} - 8 x = 3 x (3 x) + 3 x \cdot - 8 - 8 (3 x) - 8 \cdot - 8$

Step 2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$3 x^{2} - 8 x = 3 \cdot 3 x \cdot x + 3 x \cdot - 8 - 8 (3 x) - 8 \cdot - 8$

Step 2.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.3.1.2.1

Move $x$ .

$3 x^{2} - 8 x = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 8 - 8 (3 x) - 8 \cdot - 8$

Step 2.3.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} - 8 x = 3 \cdot 3 x^{2} + 3 x \cdot - 8 - 8 (3 x) - 8 \cdot - 8$

Step 2.3.1.3

Multiply $3$ by $3$ .

$3 x^{2} - 8 x = 9 x^{2} + 3 x \cdot - 8 - 8 (3 x) - 8 \cdot - 8$

Step 2.3.1.4

Multiply $- 8$ by $3$ .

$3 x^{2} - 8 x = 9 x^{2} - 24 x - 8 (3 x) - 8 \cdot - 8$

Step 2.3.1.5

Multiply $3$ by $- 8$ .

$3 x^{2} - 8 x = 9 x^{2} - 24 x - 24 x - 8 \cdot - 8$

Step 2.3.1.6

Multiply $- 8$ by $- 8$ .

$3 x^{2} - 8 x = 9 x^{2} - 24 x - 24 x + 64$

Step 2.3.2

Subtract $24 x$ from $- 24 x$ .

$3 x^{2} - 8 x = 9 x^{2} - 48 x + 64$

Step 3

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

Tap for more steps...

Step 3.1

Subtract $9 x^{2}$ from both sides of the equation.

$3 x^{2} - 8 x - 9 x^{2} = - 48 x + 64$

Step 3.2

Add $48 x$ to both sides of the equation.

$3 x^{2} - 8 x - 9 x^{2} + 48 x = 64$

Step 3.3

Subtract $9 x^{2}$ from $3 x^{2}$ .

$- 6 x^{2} - 8 x + 48 x = 64$

Step 3.4

Add $- 8 x$ and $48 x$ .

$- 6 x^{2} + 40 x = 64$

Step 4

Subtract $64$ from both sides of the equation.

$- 6 x^{2} + 40 x - 64 = 0$

Step 5

Factor the left side of the equation.

Tap for more steps...

Step 5.1

Factor $- 2$ out of $- 6 x^{2} + 40 x - 64$ .

Tap for more steps...

Step 5.1.1

Factor $- 2$ out of $- 6 x^{2}$ .

$- 2 (3 x^{2}) + 40 x - 64 = 0$

Step 5.1.2

Factor $- 2$ out of $40 x$ .

$- 2 (3 x^{2}) - 2 (- 20 x) - 64 = 0$

Step 5.1.3

Factor $- 2$ out of $- 64$ .

$- 2 (3 x^{2}) - 2 (- 20 x) - 2 \cdot 32 = 0$

Step 5.1.4

Factor $- 2$ out of $- 2 (3 x^{2}) - 2 (- 20 x)$ .

$- 2 (3 x^{2} - 20 x) - 2 \cdot 32 = 0$

Step 5.1.5

Factor $- 2$ out of $- 2 (3 x^{2} - 20 x) - 2 (32)$ .

$- 2 (3 x^{2} - 20 x + 32) = 0$

Step 5.2

Factor.

Tap for more steps...

Step 5.2.1

Factor by grouping.

Tap for more steps...

Step 5.2.1.1

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 $a \cdot c = 3 \cdot 32 = 96$ and whose sum is $b = - 20$ .

Tap for more steps...

Step 5.2.1.1.1

Factor $- 20$ out of $- 20 x$ .

$- 2 (3 x^{2} - 20 x + 32) = 0$

Step 5.2.1.1.2

Rewrite $- 20$ as $- 8$ plus $- 12$

$- 2 (3 x^{2} + (- 8 - 12) x + 32) = 0$

Step 5.2.1.1.3

Apply the distributive property.

$- 2 (3 x^{2} - 8 x - 12 x + 32) = 0$

Step 5.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.2.1.2.1

Group the first two terms and the last two terms.

$- 2 ((3 x^{2} - 8 x) - 12 x + 32) = 0$

Step 5.2.1.2.2

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

$- 2 (x (3 x - 8) - 4 (3 x - 8)) = 0$

Step 5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 8$ .

$- 2 ((3 x - 8) (x - 4)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- 2 (3 x - 8) (x - 4) = 0$

Step 6

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

$3 x - 8 = 0$

$x - 4 = 0$

Step 7

Set $3 x - 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $3 x - 8$ equal to $0$ .

$3 x - 8 = 0$

Step 7.2

Solve $3 x - 8 = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

Add $8$ to both sides of the equation.

$3 x = 8$

Step 7.2.2

Divide each term in $3 x = 8$ by $3$ and simplify.

Tap for more steps...

Step 7.2.2.1

Divide each term in $3 x = 8$ by $3$ .

$\frac{3 x}{3} = \frac{8}{3}$

Step 7.2.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{8}{3}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{3}$

Step 8

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 9

The final solution is all the values that make $- 2 (3 x - 8) (x - 4) = 0$ true.

$x = \frac{8}{3}, 4$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}, 4$

Decimal Form:

$x = 2.\overline{6}, 4$

Mixed Number Form:

$x = 2 \frac{2}{3}, 4$