Algebra Examples

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

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $2 x (x - 7)$ .

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Move $x$ .

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

Step 1.1.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.1.3

Multiply $- 7$ by $2$ .

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

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $14 x$ to both sides of the equation.

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

Step 1.3

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

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Apply the distributive property.

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

Step 1.3.1.2.2

Apply the distributive property.

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

Step 1.3.1.2.3

Apply the distributive property.

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

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.3.1.2

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

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Step 1.3.1.3.1.2.1

Move $x$ .

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

Step 1.3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 14 - 2 x^{2} + 14 x = 0$

Step 1.3.1.3.1.4

Multiply $- 3$ by $2$ .

$4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3 - 14 - 2 x^{2} + 14 x = 0$

Step 1.3.1.3.1.5

Multiply $2$ by $- 3$ .

$4 x^{2} - 6 x - 6 x - 3 \cdot - 3 - 14 - 2 x^{2} + 14 x = 0$

Step 1.3.1.3.1.6

Multiply $- 3$ by $- 3$ .

$4 x^{2} - 6 x - 6 x + 9 - 14 - 2 x^{2} + 14 x = 0$

Step 1.3.1.3.2

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

$4 x^{2} - 12 x + 9 - 14 - 2 x^{2} + 14 x = 0$

Step 1.3.2

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

$2 x^{2} - 12 x + 9 - 14 + 14 x = 0$

Step 1.3.3

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

$2 x^{2} + 2 x + 9 - 14 = 0$

Step 1.3.4

Subtract $14$ from $9$ .

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

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

Step 4.1.2.2

Multiply $- 8$ by $- 5$ .

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

Step 4.1.3

Add $4$ and $40$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Factor $4$ out of $44$ .

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

Step 4.1.4.2

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

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

Step 4.1.5

Pull terms out from under the radical.

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

Step 4.2

Multiply $2$ by $2$ .

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

Step 4.3

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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