Pre-Algebra Examples

$(2 x + 3) (2 x - 3) = 91$

Step 1

Simplify $(2 x + 3) (2 x - 3)$ .

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

$2 x (2 x - 3) + 3 (2 x - 3) = 91$

Step 1.1.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 + 3 (2 x - 3) = 91$

Step 1.1.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 3 + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 3 + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 3 + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2.1.4

Multiply $- 3$ by $2$ .

$4 x^{2} - 6 x + 3 (2 x) + 3 \cdot - 3 = 91$

Step 1.2.1.5

Multiply $2$ by $3$ .

$4 x^{2} - 6 x + 6 x + 3 \cdot - 3 = 91$

Step 1.2.1.6

Multiply $3$ by $- 3$ .

$4 x^{2} - 6 x + 6 x - 9 = 91$

Step 1.2.2

Add $- 6 x$ and $6 x$ .

$4 x^{2} + 0 - 9 = 91$

Step 1.2.3

Add $4 x^{2}$ and $0$ .

$4 x^{2} - 9 = 91$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $9$ to both sides of the equation.

$4 x^{2} = 91 + 9$

Step 2.2

Add $91$ and $9$ .

$4 x^{2} = 100$

Step 3

Divide each term in $4 x^{2} = 100$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x^{2} = 100$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{100}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{100}{4}$

Step 3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{100}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $100$ by $4$ .

$x^{2} = 25$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{25}$

Step 5

Simplify $\pm \sqrt{25}$ .

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Step 5.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 5$