Algebra Examples

${(2 x - 3)}^{2} - 14 = 2 x (x - 7)$

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.

$2 x (x - 7) = {(2 x - 3)}^{2} - 14$

Step 2

Simplify $2 x (x - 7)$ .

Tap for more steps...

Step 2.1

Rewrite.

$0 + 0 + 2 x (x - 7) = {(2 x - 3)}^{2} - 14$

Step 2.2

Simplify by adding zeros.

$2 x (x - 7) = {(2 x - 3)}^{2} - 14$

Step 2.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 7 = {(2 x - 3)}^{2} - 14$

Step 2.4

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

Tap for more steps...

Step 2.4.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 7 = {(2 x - 3)}^{2} - 14$

Step 2.4.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 7 = {(2 x - 3)}^{2} - 14$

Step 2.5

Multiply $- 7$ by $2$ .

$2 x^{2} - 14 x = {(2 x - 3)}^{2} - 14$

Step 3

Simplify ${(2 x - 3)}^{2} - 14$ .

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

$2 x^{2} - 14 x = (2 x - 3) (2 x - 3) - 14$

Step 3.1.2

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.1.2.1

Apply the distributive property.

$2 x^{2} - 14 x = 2 x (2 x - 3) - 3 (2 x - 3) - 14$

Step 3.1.2.2

Apply the distributive property.

$2 x^{2} - 14 x = 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3) - 14$

Step 3.1.2.3

Apply the distributive property.

$2 x^{2} - 14 x = 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 x^{2} - 14 x = 2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3.1.2

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

Tap for more steps...

Step 3.1.3.1.2.1

Move $x$ .

$2 x^{2} - 14 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} - 14 x = 2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3.1.3

Multiply $2$ by $2$ .

$2 x^{2} - 14 x = 4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3.1.4

Multiply $- 3$ by $2$ .

$2 x^{2} - 14 x = 4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3 - 14$

Step 3.1.3.1.5

Multiply $2$ by $- 3$ .

$2 x^{2} - 14 x = 4 x^{2} - 6 x - 6 x - 3 \cdot - 3 - 14$

Step 3.1.3.1.6

Multiply $- 3$ by $- 3$ .

$2 x^{2} - 14 x = 4 x^{2} - 6 x - 6 x + 9 - 14$

Step 3.1.3.2

Subtract $6 x$ from $- 6 x$ .

$2 x^{2} - 14 x = 4 x^{2} - 12 x + 9 - 14$

Step 3.2

Subtract $14$ from $9$ .

$2 x^{2} - 14 x = 4 x^{2} - 12 x - 5$

Step 4

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

Tap for more steps...

Step 4.1

Subtract $4 x^{2}$ from both sides of the equation.

$2 x^{2} - 14 x - 4 x^{2} = - 12 x - 5$

Step 4.2

Add $12 x$ to both sides of the equation.

$2 x^{2} - 14 x - 4 x^{2} + 12 x = - 5$

Step 4.3

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} - 14 x + 12 x = - 5$

Step 4.4

Add $- 14 x$ and $12 x$ .

$- 2 x^{2} - 2 x = - 5$

Step 5

Add $5$ to both sides of the equation.

$- 2 x^{2} - 2 x + 5 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = - 2$ , $b = - 2$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 2 \cdot 5)}}{2 \cdot - 2}$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 2 \cdot 5}}{2 \cdot - 2}$

Step 8.1.2

Multiply $- 4 \cdot - 2 \cdot 5$ .

Tap for more steps...

Step 8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{2 \pm \sqrt{4 + 8 \cdot 5}}{2 \cdot - 2}$

Step 8.1.2.2

Multiply $8$ by $5$ .

$x = \frac{2 \pm \sqrt{4 + 40}}{2 \cdot - 2}$

Step 8.1.3

Add $4$ and $40$ .

$x = \frac{2 \pm \sqrt{44}}{2 \cdot - 2}$

Step 8.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

Tap for more steps...

Step 8.1.4.1

Factor $4$ out of $44$ .

$x = \frac{2 \pm \sqrt{4 (11)}}{2 \cdot - 2}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot - 2}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{11}}{2 \cdot - 2}$

Step 8.2

Multiply $2$ by $- 2$ .

$x = \frac{2 \pm 2 \sqrt{11}}{- 4}$

Step 8.3

Simplify $\frac{2 \pm 2 \sqrt{11}}{- 4}$ .

$x = \frac{1 \pm \sqrt{11}}{- 2}$

Step 8.4

Move the negative in front of the fraction.

$x = - \frac{1 \pm \sqrt{11}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{1 + \sqrt{11}}{2}, - \frac{1 - \sqrt{11}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 + \sqrt{11}}{2}, - \frac{1 - \sqrt{11}}{2}$

Decimal Form:

$x = - 2.15831239 \dots, 1.15831239 \dots$