Finite Math Examples

$\sqrt{2 y} + 6 + 7 = y + 6$

Step 1

Solve for $\sqrt{2 y}$ .

Tap for more steps...

Step 1.1

Add $6$ and $7$ .

$\sqrt{2 y} + 13 = y + 6$

Step 1.2

Move all terms not containing $\sqrt{2 y}$ to the right side of the equation.

Tap for more steps...

Step 1.2.1

Subtract $13$ from both sides of the equation.

$\sqrt{2 y} = y + 6 - 13$

Step 1.2.2

Subtract $13$ from $6$ .

$\sqrt{2 y} = y - 7$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 y}}^{2} = {(y - 7)}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y}$ as ${(2 y)}^{\frac{1}{2}}$ .

${({(2 y)}^{\frac{1}{2}})}^{2} = {(y - 7)}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(2 y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(2 y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 y)}^{\frac{1}{2} \cdot 2} = {(y - 7)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(2 y)}^{\frac{1}{2} \cdot 2} = {(y - 7)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 y)}^{1} = {(y - 7)}^{2}$

Step 3.2.1.2

Simplify.

$2 y = {(y - 7)}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(y - 7)}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Rewrite ${(y - 7)}^{2}$ as $(y - 7) (y - 7)$ .

$2 y = (y - 7) (y - 7)$

Step 3.3.1.2

Expand $(y - 7) (y - 7)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$2 y = y (y - 7) - 7 (y - 7)$

Step 3.3.1.2.2

Apply the distributive property.

$2 y = y \cdot y + y \cdot - 7 - 7 (y - 7)$

Step 3.3.1.2.3

Apply the distributive property.

$2 y = y \cdot y + y \cdot - 7 - 7 y - 7 \cdot - 7$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Multiply $y$ by $y$ .

$2 y = y^{2} + y \cdot - 7 - 7 y - 7 \cdot - 7$

Step 3.3.1.3.1.2

Move $- 7$ to the left of $y$ .

$2 y = y^{2} - 7 \cdot y - 7 y - 7 \cdot - 7$

Step 3.3.1.3.1.3

Multiply $- 7$ by $- 7$ .

$2 y = y^{2} - 7 y - 7 y + 49$

Step 3.3.1.3.2

Subtract $7 y$ from $- 7 y$ .

$2 y = y^{2} - 14 y + 49$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y^{2} - 14 y + 49 = 2 y$

Step 4.2

Move all terms containing $y$ to the left side of the equation.

Tap for more steps...

Step 4.2.1

Subtract $2 y$ from both sides of the equation.

$y^{2} - 14 y + 49 - 2 y = 0$

Step 4.2.2

Subtract $2 y$ from $- 14 y$ .

$y^{2} - 16 y + 49 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 1$ , $b = - 16$ , and $c = 49$ into the quadratic formula and solve for $y$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 49)}}{2 \cdot 1}$

Step 4.5

Simplify.

Tap for more steps...

Step 4.5.1

Simplify the numerator.

Tap for more steps...

Step 4.5.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 49$ .

Tap for more steps...

Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 49}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $49$ .

$y = \frac{16 \pm \sqrt{256 - 196}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $196$ from $256$ .

$y = \frac{16 \pm \sqrt{60}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

Tap for more steps...

Step 4.5.1.4.1

Factor $4$ out of $60$ .

$y = \frac{16 \pm \sqrt{4 (15)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$y = \frac{16 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$y = \frac{16 \pm 2 \sqrt{15}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$y = \frac{16 \pm 2 \sqrt{15}}{2}$

Step 4.5.3

Simplify $\frac{16 \pm 2 \sqrt{15}}{2}$ .

$y = 8 \pm \sqrt{15}$

Step 4.6

The final answer is the combination of both solutions.

$y = 8 + \sqrt{15}, 8 - \sqrt{15}$

Step 5

Exclude the solutions that do not make $\sqrt{2 y} + 6 + 7 = y + 6$ true.

$y = 8 + \sqrt{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = 8 + \sqrt{15}$

Decimal Form:

$y = 11.87298334 \dots$