Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$x^{2} + (- 2 a - 2 \cdot 1) x + 3 a + 2 = 0$

Step 1.2

Multiply $- 2$ by $1$ .

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

Step 1.3

Apply the distributive property.

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

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $- 2$ by $- 1$ .

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

Step 4.1.3

Multiply $- 1$ by $- 2$ .

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

Step 4.1.4

Add parentheses.

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

Step 4.1.5

Let $u = 1 \cdot (3 a + 2)$ . Substitute $u$ for all occurrences of $1 \cdot (3 a + 2)$ .

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Step 4.1.5.1

Rewrite ${(- 2 a - 2)}^{2}$ as $(- 2 a - 2) (- 2 a - 2)$ .

$x = \frac{2 a + 2 \pm \sqrt{(- 2 a - 2) (- 2 a - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.2

Expand $(- 2 a - 2) (- 2 a - 2)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$x = \frac{2 a + 2 \pm \sqrt{- 2 a (- 2 a - 2) - 2 (- 2 a - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.2.2

Apply the distributive property.

$x = \frac{2 a + 2 \pm \sqrt{- 2 a (- 2 a) - 2 a \cdot - 2 - 2 (- 2 a - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.2.3

Apply the distributive property.

$x = \frac{2 a + 2 \pm \sqrt{- 2 a (- 2 a) - 2 a \cdot - 2 - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3

Simplify and combine like terms.

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Step 4.1.5.3.1

Simplify each term.

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Step 4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 a + 2 \pm \sqrt{- 2 \cdot (- 2 a \cdot a) - 2 a \cdot - 2 - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.5.3.1.2.1

Move $a$ .

$x = \frac{2 a + 2 \pm \sqrt{- 2 \cdot (- 2 (a \cdot a)) - 2 a \cdot - 2 - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.2.2

Multiply $a$ by $a$ .

$x = \frac{2 a + 2 \pm \sqrt{- 2 \cdot (- 2 a^{2}) - 2 a \cdot - 2 - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} - 2 a \cdot - 2 - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.4

Multiply $- 2$ by $- 2$ .

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} + 4 a - 2 (- 2 a) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.5

Multiply $- 2$ by $- 2$ .

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} + 4 a + 4 a - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.1.6

Multiply $- 2$ by $- 2$ .

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} + 4 a + 4 a + 4 - 4 \cdot u}}{2 \cdot 1}$

Step 4.1.5.3.2

Add $4 a$ and $4 a$ .

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} + 8 a + 4 - 4 \cdot u}}{2 \cdot 1}$

$x = \frac{2 a + 2 \pm \sqrt{4 a^{2} + 8 a + 4 - 4 u}}{2 \cdot 1}$

Step 4.1.6

Factor $4$ out of $4 a^{2} + 8 a + 4 - 4 u$ .

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Step 4.1.6.1

Factor $4$ out of $4 a^{2}$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2}) + 8 a + 4 - 4 u}}{2 \cdot 1}$

Step 4.1.6.2

Factor $4$ out of $8 a$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2}) + 4 (2 a) + 4 - 4 u}}{2 \cdot 1}$

Step 4.1.6.3

Factor $4$ out of $4$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2}) + 4 (2 a) + 4 (1) - 4 u}}{2 \cdot 1}$

Step 4.1.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2}) + 4 (2 a) + 4 (1) + 4 (- u)}}{2 \cdot 1}$

Step 4.1.6.5

Factor $4$ out of $4 (a^{2}) + 4 (2 a)$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2} + 2 a) + 4 (1) + 4 (- u)}}{2 \cdot 1}$

Step 4.1.6.6

Factor $4$ out of $4 (a^{2} + 2 a) + 4 (1)$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2} + 2 a + 1) + 4 (- u)}}{2 \cdot 1}$

Step 4.1.6.7

Factor $4$ out of $4 (a^{2} + 2 a + 1) + 4 (- u)$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2} + 2 a + 1 - u)}}{2 \cdot 1}$

Step 4.1.7

Replace all occurrences of $u$ with $1 \cdot (3 a + 2)$ .

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

Step 4.1.8

Simplify.

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Step 4.1.8.1

Simplify each term.

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Step 4.1.8.1.1

Multiply $3 a + 2$ by $1$ .

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

Step 4.1.8.1.2

Apply the distributive property.

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

Step 4.1.8.1.3

Multiply $3$ by $- 1$ .

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

Step 4.1.8.1.4

Multiply $- 1$ by $2$ .

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

Step 4.1.8.2

Subtract $3 a$ from $2 a$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2} - a + 1 - 2)}}{2 \cdot 1}$

Step 4.1.8.3

Subtract $2$ from $1$ .

$x = \frac{2 a + 2 \pm \sqrt{4 (a^{2} - a - 1)}}{2 \cdot 1}$

Step 4.1.9

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

$x = \frac{2 a + 2 \pm \sqrt{2^{2} (a^{2} - a - 1)}}{2 \cdot 1}$

Step 4.1.10

Pull terms out from under the radical.

$x = \frac{2 a + 2 \pm 2 \sqrt{a^{2} - a - 1}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{2 a + 2 \pm 2 \sqrt{a^{2} - a - 1}}{2}$

Step 5

The final answer is the combination of both solutions.

$x = a + 1 + \sqrt{a^{2} - a - 1}$

$x = a + 1 - \sqrt{a^{2} - a - 1}$