Precalculus Examples

$(x + 7) (2 x + 1) = 7$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $(x + 7) (2 x + 1)$ .

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Step 1.1.1.1

Expand $(x + 7) (2 x + 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$x (2 x + 1) + 7 (2 x + 1) = 7$

Step 1.1.1.1.2

Apply the distributive property.

$x (2 x) + x \cdot 1 + 7 (2 x + 1) = 7$

Step 1.1.1.1.3

Apply the distributive property.

$x (2 x) + x \cdot 1 + 7 (2 x) + 7 \cdot 1 = 7$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot 1 + 7 (2 x) + 7 \cdot 1 = 7$

Step 1.1.1.2.1.2

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

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Step 1.1.1.2.1.2.1

Move $x$ .

$2 (x \cdot x) + x \cdot 1 + 7 (2 x) + 7 \cdot 1 = 7$

Step 1.1.1.2.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + x \cdot 1 + 7 (2 x) + 7 \cdot 1 = 7$

Step 1.1.1.2.1.3

Multiply $x$ by $1$ .

$2 x^{2} + x + 7 (2 x) + 7 \cdot 1 = 7$

Step 1.1.1.2.1.4

Multiply $2$ by $7$ .

$2 x^{2} + x + 14 x + 7 \cdot 1 = 7$

Step 1.1.1.2.1.5

Multiply $7$ by $1$ .

$2 x^{2} + x + 14 x + 7 = 7$

Step 1.1.1.2.2

Add $x$ and $14 x$ .

$2 x^{2} + 15 x + 7 = 7$

Step 1.2

Subtract $7$ from both sides of the equation.

$2 x^{2} + 15 x + 7 - 7 = 0$

Step 1.3

Combine the opposite terms in $2 x^{2} + 15 x + 7 - 7$ .

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Step 1.3.1

Subtract $7$ from $7$ .

$2 x^{2} + 15 x + 0 = 0$

Step 1.3.2

Add $2 x^{2} + 15 x$ and $0$ .

$2 x^{2} + 15 x = 0$

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 $a = 2$ , $b = 15$ , and $c = 0$ into the quadratic formula and solve for $x$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (2 \cdot 0)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot 2 \cdot 0}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 0$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 8 \cdot 0}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $0$ .

$x = \frac{- 15 \pm \sqrt{225 + 0}}{2 \cdot 2}$

Step 4.1.3

Add $225$ and $0$ .

$x = \frac{- 15 \pm \sqrt{225}}{2 \cdot 2}$

Step 4.1.4

Rewrite $225$ as $15^{2}$ .

$x = \frac{- 15 \pm \sqrt{15^{2}}}{2 \cdot 2}$

Step 4.1.5

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

$x = \frac{- 15 \pm 15}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 15 \pm 15}{4}$

Step 5

The final answer is the combination of both solutions.

$x = 0, - \frac{15}{2}$