Basic Math Examples

$a^{2} + y^{2} + 2 a + 2 y - 7 = 0$

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 $a = 1$ , $b = 2$ , and $c = a^{2} + 2 a - 7$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot (a^{2} + 2 a - 7))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot (a^{2} + 2 a - 7)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot (a^{2} + 2 a - 7)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 - 4 a^{2} - 4 (2 a) - 4 \cdot - 7}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $2$ by $- 4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 a^{2} - 8 a - 4 \cdot - 7}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 4$ by $- 7$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 a^{2} - 8 a + 28}}{2 \cdot 1}$

Step 3.1.5

Add $4$ and $28$ .

$y = \frac{- 2 \pm \sqrt{- 4 a^{2} - 8 a + 32}}{2 \cdot 1}$

Step 3.1.6

Rewrite $- 4 a^{2} - 8 a + 32$ in a factored form.

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Step 3.1.6.1

Factor $4$ out of $- 4 a^{2} - 8 a + 32$ .

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Step 3.1.6.1.1

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

$y = \frac{- 2 \pm \sqrt{4 (- a^{2}) - 8 a + 32}}{2 \cdot 1}$

Step 3.1.6.1.2

Factor $4$ out of $- 8 a$ .

$y = \frac{- 2 \pm \sqrt{4 (- a^{2}) + 4 (- 2 a) + 32}}{2 \cdot 1}$

Step 3.1.6.1.3

Factor $4$ out of $32$ .

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

Step 3.1.6.1.4

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

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

Step 3.1.6.1.5

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

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

Step 3.1.6.2

Factor by grouping.

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Step 3.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 3.1.6.2.1.1

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

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

Step 3.1.6.2.1.2

Rewrite $- 2$ as $2$ plus $- 4$

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

Step 3.1.6.2.1.3

Apply the distributive property.

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

Step 3.1.6.2.2

Factor out the greatest common factor from each group.

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Step 3.1.6.2.2.1

Group the first two terms and the last two terms.

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

Step 3.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- a + 2$ .

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

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

Step 3.1.7

Rewrite $4 (- a + 2) (a + 4)$ as $2^{2} ((- a + 2) (a + 2^{2}))$ .

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Step 3.1.7.1

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

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

Step 3.1.7.2

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

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

Step 3.1.7.3

Add parentheses.

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

Step 3.1.8

Pull terms out from under the radical.

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

Step 3.1.9

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

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

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- a + 2) (a + 4)}}{2}$

Step 3.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- a + 2) (a + 4)}}{2}$ .

$y = - 1 \pm \sqrt{(- a + 2) (a + 4)}$

Step 4

The final answer is the combination of both solutions.

$y = - 1 + \sqrt{(- a + 2) (a + 4)}$

$y = - 1 - \sqrt{(- a + 2) (a + 4)}$