Algebra Examples

$(2 x - 1) (2 x + 1) = x (2 x + 3)$

Step 1

Simplify $(2 x - 1) (2 x + 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + (2 x - 1) (2 x + 1) = x (2 x + 3)$

Step 1.2

Simplify by adding zeros.

$(2 x - 1) (2 x + 1) = x (2 x + 3)$

Step 1.3

Expand $(2 x - 1) (2 x + 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 x (2 x + 1) - 1 (2 x + 1) = x (2 x + 3)$

Step 1.3.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 1 - 1 (2 x + 1) = x (2 x + 3)$

Step 1.3.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 1 - 1 (2 x) - 1 \cdot 1 = x (2 x + 3)$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine the opposite terms in $2 x (2 x) + 2 x \cdot 1 - 1 (2 x) - 1 \cdot 1$ .

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Step 1.4.1.1

Reorder the factors in the terms $2 x \cdot 1$ and $- 1 (2 x)$ .

$2 x (2 x) + 1 \cdot 2 x - 1 \cdot 2 x - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.1.2

Subtract $2 x$ from $1 \cdot 2 x$ .

$2 x (2 x) + 0 - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.1.3

Add $2 x (2 x)$ and $0$ .

$2 x (2 x) - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.2.2

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

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Step 1.4.2.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.2.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.2.3

Multiply $2$ by $2$ .

$4 x^{2} - 1 \cdot 1 = x (2 x + 3)$

Step 1.4.2.4

Multiply $- 1$ by $1$ .

$4 x^{2} - 1 = x (2 x + 3)$

Step 2

Simplify $x (2 x + 3)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$4 x^{2} - 1 = x (2 x) + x \cdot 3$

Step 2.1.2

Reorder.

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Step 2.1.2.1

Rewrite using the commutative property of multiplication.

$4 x^{2} - 1 = 2 x \cdot x + x \cdot 3$

Step 2.1.2.2

Move $3$ to the left of $x$ .

$4 x^{2} - 1 = 2 x \cdot x + 3 \cdot x$

Step 2.2

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

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Step 2.2.1

Move $x$ .

$4 x^{2} - 1 = 2 (x \cdot x) + 3 \cdot x$

Step 2.2.2

Multiply $x$ by $x$ .

$4 x^{2} - 1 = 2 x^{2} + 3 \cdot x$

$4 x^{2} - 1 = 2 x^{2} + 3 x$

Step 3

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

$2 x^{2} + 3 x = 4 x^{2} - 1$

Step 4

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

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Step 4.1

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

$2 x^{2} + 3 x - 4 x^{2} = - 1$

Step 4.2

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

$- 2 x^{2} + 3 x = - 1$

Step 5

Add $1$ to both sides of the equation.

$- 2 x^{2} + 3 x + 1 = 0$

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 $a = - 2$ , $b = 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 2 \cdot 1)}}{2 \cdot - 2}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 8.1.2

Multiply $- 4 \cdot - 2 \cdot 1$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 + 8 \cdot 1}}{2 \cdot - 2}$

Step 8.1.2.2

Multiply $8$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 + 8}}{2 \cdot - 2}$

Step 8.1.3

Add $9$ and $8$ .

$x = \frac{- 3 \pm \sqrt{17}}{2 \cdot - 2}$

Step 8.2

Multiply $2$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{17}}{- 4}$

Step 8.3

Simplify $\frac{- 3 \pm \sqrt{17}}{- 4}$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Decimal Form:

$x = 1.78077640 \dots, - 0.28077640 \dots$