Precalculus Examples

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

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = 2 a$ , and $c = a^{2}$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4 \cdot 1 \cdot a^{2}$ as ${(2 \cdot 1 a)}^{2}$ .

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

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = 2 \cdot 1 a$ .

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

Step 3.1.3

Simplify.

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Step 3.1.3.1

Factor $2 a$ out of $2 a + 2 \cdot 1 a$ .

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Step 3.1.3.1.1

Factor $2 a$ out of $2 a$ .

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

Step 3.1.3.1.2

Factor $2 a$ out of $2 \cdot 1 a$ .

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

Step 3.1.3.1.3

Factor $2 a$ out of $2 a (1) + 2 a (1)$ .

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

Step 3.1.3.2

Add $1$ and $1$ .

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

Step 3.1.3.3

Multiply $2$ by $2$ .

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

Step 3.1.3.4

Combine exponents.

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Step 3.1.3.4.1

Multiply $2$ by $1$ .

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

Step 3.1.3.4.2

Multiply $2$ by $- 1$ .

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

Step 3.1.4

Subtract $2 a$ from $2 a$ .

$x = \frac{- 2 a \pm \sqrt{4 a \cdot 0}}{2 \cdot 1}$

Step 3.1.5

Combine exponents.

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Step 3.1.5.1

Multiply $0$ by $4$ .

$x = \frac{- 2 a \pm \sqrt{0 a}}{2 \cdot 1}$

Step 3.1.5.2

Multiply $0$ by $a$ .

$x = \frac{- 2 a \pm \sqrt{0}}{2 \cdot 1}$

Step 3.1.6

Rewrite $0$ as $0^{2}$ .

$x = \frac{- 2 a \pm \sqrt{0^{2}}}{2 \cdot 1}$

Step 3.1.7

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

$x = \frac{- 2 a \pm 0}{2 \cdot 1}$

Step 3.1.8

$- 2 a$ plus or minus $0$ is $- 2 a$ .

$x = \frac{- 2 a}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 a}{2}$

Step 3.3

Cancel the common factor of $- 2$ and $2$ .

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Step 3.3.1

Factor $2$ out of $- 2 a$ .

$x = \frac{2 (- a)}{2}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- a)}{2 (1)}$

Step 3.3.2.2

Cancel the common factor.

$x = \frac{2 (- a)}{2 \cdot 1}$

Step 3.3.2.3

Rewrite the expression.

$x = \frac{- a}{1}$

Step 3.3.2.4

Divide $- a$ by $1$ .

$x = - a$

Step 4

The final answer is the combination of both solutions.

$x = - a$ Double roots