Pre-Algebra Examples

$y^{2} - 6 y + 7 = 0$

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 $a = 1$ , $b = - 6$ , and $c = 7$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 7}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $7$ .

$y = \frac{6 \pm \sqrt{36 - 28}}{2 \cdot 1}$

Step 3.1.3

Subtract $28$ from $36$ .

$y = \frac{6 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.4.1

Factor $4$ out of $8$ .

$y = \frac{6 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 3.1.4.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{6 \pm 2 \sqrt{2}}{2}$

Step 3.3

Simplify $\frac{6 \pm 2 \sqrt{2}}{2}$ .

$y = 3 \pm \sqrt{2}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 7}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $7$ .

$y = \frac{6 \pm \sqrt{36 - 28}}{2 \cdot 1}$

Step 4.1.3

Subtract $28$ from $36$ .

$y = \frac{6 \pm \sqrt{8}}{2 \cdot 1}$

Step 4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.4.1

Factor $4$ out of $8$ .

$y = \frac{6 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 4.1.4.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{6 \pm 2 \sqrt{2}}{2}$

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{2}}{2}$ .

$y = 3 \pm \sqrt{2}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 3 + \sqrt{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 7}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $7$ .

$y = \frac{6 \pm \sqrt{36 - 28}}{2 \cdot 1}$

Step 5.1.3

Subtract $28$ from $36$ .

$y = \frac{6 \pm \sqrt{8}}{2 \cdot 1}$

Step 5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $4$ out of $8$ .

$y = \frac{6 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 5.1.4.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{6 \pm 2 \sqrt{2}}{2}$

Step 5.3

Simplify $\frac{6 \pm 2 \sqrt{2}}{2}$ .

$y = 3 \pm \sqrt{2}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 3 - \sqrt{2}$

Step 6

The final answer is the combination of both solutions.

$y = 3 + \sqrt{2}, 3 - \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 3 + \sqrt{2}, 3 - \sqrt{2}$

Decimal Form:

$y = 4.41421356 \dots, 1.58578643 \dots$