Finite Math Examples

$y = x^{2} + 2 x - 3$ , $y = 8 - 2 x - x^{2}$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} + 2 x - 3 = 8 - 2 x - x^{2}$

Step 2

Solve $x^{2} + 2 x - 3 = 8 - 2 x - x^{2}$ for $x$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Add $2 x$ to both sides of the equation.

$x^{2} + 2 x - 3 + 2 x = 8 - x^{2}$

Step 2.1.2

Add $x^{2}$ to both sides of the equation.

$x^{2} + 2 x - 3 + 2 x + x^{2} = 8$

Step 2.1.3

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 2 x - 3 + 2 x = 8$

Step 2.1.4

Add $2 x$ and $2 x$ .

$2 x^{2} + 4 x - 3 = 8$

Step 2.2

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 2.2.1

Subtract $8$ from both sides of the equation.

$2 x^{2} + 4 x - 3 - 8 = 0$

Step 2.2.2

Subtract $8$ from $- 3$ .

$2 x^{2} + 4 x - 11 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 8 - 2 x - x^{2}$

Step 2.4

Substitute the values $a = 2$ , $b = 4$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (2 \cdot - 11)}}{2 \cdot 2} + y = 8 - 2 x - x^{2}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot - 11}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 11$ .

Tap for more steps...

Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot - 11}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $- 11$ .

$x = \frac{- 4 \pm \sqrt{16 + 88}}{2 \cdot 2}$

Step 2.5.1.3

Add $16$ and $88$ .

$x = \frac{- 4 \pm \sqrt{104}}{2 \cdot 2}$

Step 2.5.1.4

Rewrite $104$ as $2^{2} \cdot 26$ .

Tap for more steps...

Step 2.5.1.4.1

Factor $4$ out of $104$ .

$x = \frac{- 4 \pm \sqrt{4 (26)}}{2 \cdot 2}$

Step 2.5.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 2}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{26}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 \sqrt{26}}{4}$

Step 2.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{26}}{4}$ .

$x = \frac{- 2 \pm \sqrt{26}}{2}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

Simplify the numerator.

Tap for more steps...

Step 2.6.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot - 11}}{2 \cdot 2}$

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot - 11$ .

Tap for more steps...

Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot - 11}}{2 \cdot 2}$

Step 2.6.1.2.2

Multiply $- 8$ by $- 11$ .

$x = \frac{- 4 \pm \sqrt{16 + 88}}{2 \cdot 2}$

Step 2.6.1.3

Add $16$ and $88$ .

$x = \frac{- 4 \pm \sqrt{104}}{2 \cdot 2}$

Step 2.6.1.4

Rewrite $104$ as $2^{2} \cdot 26$ .

Tap for more steps...

Step 2.6.1.4.1

Factor $4$ out of $104$ .

$x = \frac{- 4 \pm \sqrt{4 (26)}}{2 \cdot 2}$

Step 2.6.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 2}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{26}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 \sqrt{26}}{4}$

Step 2.6.3

Simplify $\frac{- 4 \pm 2 \sqrt{26}}{4}$ .

$x = \frac{- 2 \pm \sqrt{26}}{2}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + \sqrt{26}}{2}$

Step 2.6.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + \sqrt{26}}{2}$

Step 2.6.6

Factor $- 1$ out of $\sqrt{26}$ .

$x = \frac{- 1 \cdot 2 - 1 (- \sqrt{26})}{2}$

Step 2.6.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{26})$ .

$x = \frac{- 1 (2 - \sqrt{26})}{2}$

Step 2.6.8

Move the negative in front of the fraction.

$x = - \frac{2 - \sqrt{26}}{2}$

Step 2.7

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

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

Step 2.7.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot - 11}}{2 \cdot 2}$

Step 2.7.1.2

Multiply $- 4 \cdot 2 \cdot - 11$ .

Tap for more steps...

Step 2.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot - 11}}{2 \cdot 2}$

Step 2.7.1.2.2

Multiply $- 8$ by $- 11$ .

$x = \frac{- 4 \pm \sqrt{16 + 88}}{2 \cdot 2}$

Step 2.7.1.3

Add $16$ and $88$ .

$x = \frac{- 4 \pm \sqrt{104}}{2 \cdot 2}$

Step 2.7.1.4

Rewrite $104$ as $2^{2} \cdot 26$ .

Tap for more steps...

Step 2.7.1.4.1

Factor $4$ out of $104$ .

$x = \frac{- 4 \pm \sqrt{4 (26)}}{2 \cdot 2}$

Step 2.7.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 2}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{26}}{2 \cdot 2}$

Step 2.7.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 \sqrt{26}}{4}$

Step 2.7.3

Simplify $\frac{- 4 \pm 2 \sqrt{26}}{4}$ .

$x = \frac{- 2 \pm \sqrt{26}}{2}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - \sqrt{26}}{2}$

Step 2.7.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - \sqrt{26}}{2}$

Step 2.7.6

Factor $- 1$ out of $- \sqrt{26}$ .

$x = \frac{- 1 \cdot 2 - (\sqrt{26})}{2}$

Step 2.7.7

Factor $- 1$ out of $- 1 (2) - (\sqrt{26})$ .

$x = \frac{- 1 (2 + \sqrt{26})}{2}$

Step 2.7.8

Move the negative in front of the fraction.

$x = - \frac{2 + \sqrt{26}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = - \frac{2 - \sqrt{26}}{2}, - \frac{2 + \sqrt{26}}{2}$

Step 3

Evaluate $y$ when $x = - \frac{2 - \sqrt{26}}{2}$ .

Tap for more steps...

Step 3.1

Substitute $- \frac{2 - \sqrt{26}}{2}$ for $x$ .

$y = 8 - 2 (- \frac{2 - \sqrt{26}}{2}) - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2

Simplify $8 - 2 (- \frac{2 - \sqrt{26}}{2}) - {(- \frac{2 - \sqrt{26}}{2})}^{2}$ .

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.1

Move the leading negative in $- \frac{2 - \sqrt{26}}{2}$ into the numerator.

$y = 8 - 2 \frac{- (2 - \sqrt{26})}{2} - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.1.2

Factor $2$ out of $- 2$ .

$y = 8 + 2 (- 1) \frac{- (2 - \sqrt{26})}{2} - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.1.3

Cancel the common factor.

$y = 8 + 2 \cdot - 1 \frac{- (2 - \sqrt{26})}{2} - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.1.4

Rewrite the expression.

$y = 8 - 1 (- (2 - \sqrt{26})) - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.2

Multiply $- 1$ by $- 1$ .

$y = 8 + 1 (2 - \sqrt{26}) - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.3

Multiply $2 - \sqrt{26}$ by $1$ .

$y = 8 + 2 - \sqrt{26} - {(- \frac{2 - \sqrt{26}}{2})}^{2}$

Step 3.2.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.2.1.4.1

Apply the product rule to $- \frac{2 - \sqrt{26}}{2}$ .

$y = 8 + 2 - \sqrt{26} - ({(- 1)}^{2} {(\frac{2 - \sqrt{26}}{2})}^{2})$

Step 3.2.1.4.2

Apply the product rule to $\frac{2 - \sqrt{26}}{2}$ .

$y = 8 + 2 - \sqrt{26} - ({(- 1)}^{2} \frac{{(2 - \sqrt{26})}^{2}}{2^{2}})$

Step 3.2.1.5

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 3.2.1.5.1

Move ${(- 1)}^{2}$ .

$y = 8 + 2 - \sqrt{26} + {(- 1)}^{2} \cdot - 1 \frac{{(2 - \sqrt{26})}^{2}}{2^{2}}$

Step 3.2.1.5.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

Tap for more steps...

Step 3.2.1.5.2.1

Raise $- 1$ to the power of $1$ .

$y = 8 + 2 - \sqrt{26} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{{(2 - \sqrt{26})}^{2}}{2^{2}}$

Step 3.2.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 8 + 2 - \sqrt{26} + {(- 1)}^{2 + 1} \frac{{(2 - \sqrt{26})}^{2}}{2^{2}}$

Step 3.2.1.5.3

Add $2$ and $1$ .

$y = 8 + 2 - \sqrt{26} + {(- 1)}^{3} \frac{{(2 - \sqrt{26})}^{2}}{2^{2}}$

Step 3.2.1.6

Raise $- 1$ to the power of $3$ .

$y = 8 + 2 - \sqrt{26} - \frac{{(2 - \sqrt{26})}^{2}}{2^{2}}$

Step 3.2.1.7

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

$y = 8 + 2 - \sqrt{26} - \frac{{(2 - \sqrt{26})}^{2}}{4}$

Step 3.2.1.8

Rewrite ${(2 - \sqrt{26})}^{2}$ as $(2 - \sqrt{26}) (2 - \sqrt{26})$ .

$y = 8 + 2 - \sqrt{26} - \frac{(2 - \sqrt{26}) (2 - \sqrt{26})}{4}$

Step 3.2.1.9

Expand $(2 - \sqrt{26}) (2 - \sqrt{26})$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.9.1

Apply the distributive property.

$y = 8 + 2 - \sqrt{26} - \frac{2 (2 - \sqrt{26}) - \sqrt{26} (2 - \sqrt{26})}{4}$

Step 3.2.1.9.2

Apply the distributive property.

$y = 8 + 2 - \sqrt{26} - \frac{2 \cdot 2 + 2 (- \sqrt{26}) - \sqrt{26} (2 - \sqrt{26})}{4}$

Step 3.2.1.9.3

Apply the distributive property.

$y = 8 + 2 - \sqrt{26} - \frac{2 \cdot 2 + 2 (- \sqrt{26}) - \sqrt{26} \cdot 2 - \sqrt{26} (- \sqrt{26})}{4}$

Step 3.2.1.10

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.10.1

Simplify each term.

Tap for more steps...

Step 3.2.1.10.1.1

Multiply $2$ by $2$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 + 2 (- \sqrt{26}) - \sqrt{26} \cdot 2 - \sqrt{26} (- \sqrt{26})}{4}$

Step 3.2.1.10.1.2

Multiply $- 1$ by $2$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - \sqrt{26} \cdot 2 - \sqrt{26} (- \sqrt{26})}{4}$

Step 3.2.1.10.1.3

Multiply $2$ by $- 1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} - \sqrt{26} (- \sqrt{26})}{4}$

Step 3.2.1.10.1.4

Multiply $- \sqrt{26} (- \sqrt{26})$ .

Tap for more steps...

Step 3.2.1.10.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 1 \sqrt{26} \sqrt{26}}{4}$

Step 3.2.1.10.1.4.2

Multiply $\sqrt{26}$ by $1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + \sqrt{26} \sqrt{26}}{4}$

Step 3.2.1.10.1.4.3

Raise $\sqrt{26}$ to the power of $1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + {\sqrt{26}}^{1} \sqrt{26}}{4}$

Step 3.2.1.10.1.4.4

Raise $\sqrt{26}$ to the power of $1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + {\sqrt{26}}^{1} {\sqrt{26}}^{1}}{4}$

Step 3.2.1.10.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + {\sqrt{26}}^{1 + 1}}{4}$

Step 3.2.1.10.1.4.6

Add $1$ and $1$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + {\sqrt{26}}^{2}}{4}$

Step 3.2.1.10.1.5

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

Tap for more steps...

Step 3.2.1.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + {(26^{\frac{1}{2}})}^{2}}{4}$

Step 3.2.1.10.1.5.2

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

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 26^{\frac{1}{2} \cdot 2}}{4}$

Step 3.2.1.10.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 26^{\frac{2}{2}}}{4}$

Step 3.2.1.10.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.10.1.5.4.1

Cancel the common factor.

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 26^{\frac{2}{2}}}{4}$

Step 3.2.1.10.1.5.4.2

Rewrite the expression.

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 26^{1}}{4}$

Step 3.2.1.10.1.5.5

Evaluate the exponent.

$y = 8 + 2 - \sqrt{26} - \frac{4 - 2 \sqrt{26} - 2 \sqrt{26} + 26}{4}$

Step 3.2.1.10.2

Add $4$ and $26$ .

$y = 8 + 2 - \sqrt{26} - \frac{30 - 2 \sqrt{26} - 2 \sqrt{26}}{4}$

Step 3.2.1.10.3

Subtract $2 \sqrt{26}$ from $- 2 \sqrt{26}$ .

$y = 8 + 2 - \sqrt{26} - \frac{30 - 4 \sqrt{26}}{4}$

Step 3.2.1.11

Cancel the common factor of $30 - 4 \sqrt{26}$ and $4$ .

Tap for more steps...

Step 3.2.1.11.1

Factor $2$ out of $30$ .

$y = 8 + 2 - \sqrt{26} - \frac{2 (15) - 4 \sqrt{26}}{4}$

Step 3.2.1.11.2

Factor $2$ out of $- 4 \sqrt{26}$ .

$y = 8 + 2 - \sqrt{26} - \frac{2 (15) + 2 (- 2 \sqrt{26})}{4}$

Step 3.2.1.11.3

Factor $2$ out of $2 (15) + 2 (- 2 \sqrt{26})$ .

$y = 8 + 2 - \sqrt{26} - \frac{2 (15 - 2 \sqrt{26})}{4}$

Step 3.2.1.11.4

Cancel the common factors.

Tap for more steps...

Step 3.2.1.11.4.1

Factor $2$ out of $4$ .

$y = 8 + 2 - \sqrt{26} - \frac{2 (15 - 2 \sqrt{26})}{2 \cdot 2}$

Step 3.2.1.11.4.2

Cancel the common factor.

$y = 8 + 2 - \sqrt{26} - \frac{2 (15 - 2 \sqrt{26})}{2 \cdot 2}$

Step 3.2.1.11.4.3

Rewrite the expression.

$y = 8 + 2 - \sqrt{26} - \frac{15 - 2 \sqrt{26}}{2}$

Step 3.2.2

Add $8$ and $2$ .

$y = 10 - \sqrt{26} - \frac{15 - 2 \sqrt{26}}{2}$

Step 3.2.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 10 \cdot \frac{2}{2} - \frac{15 - 2 \sqrt{26}}{2} - \sqrt{26}$

Step 3.2.4

Combine fractions.

Tap for more steps...

Step 3.2.4.1

Combine $10$ and $\frac{2}{2}$ .

$y = \frac{10 \cdot 2}{2} - \frac{15 - 2 \sqrt{26}}{2} - \sqrt{26}$

Step 3.2.4.2

Simplify the expression.

Tap for more steps...

Step 3.2.4.2.1

Combine the numerators over the common denominator.

$y = \frac{10 \cdot 2 - (15 - 2 \sqrt{26})}{2} - \sqrt{26}$

Step 3.2.4.2.2

Multiply $10$ by $2$ .

$y = \frac{20 - (15 - 2 \sqrt{26})}{2} - \sqrt{26}$

Step 3.2.5

Simplify the numerator.

Tap for more steps...

Step 3.2.5.1

Apply the distributive property.

$y = \frac{20 - 1 \cdot 15 - (- 2 \sqrt{26})}{2} - \sqrt{26}$

Step 3.2.5.2

Multiply $- 1$ by $15$ .

$y = \frac{20 - 15 - (- 2 \sqrt{26})}{2} - \sqrt{26}$

Step 3.2.5.3

Multiply $- 2$ by $- 1$ .

$y = \frac{20 - 15 + 2 \sqrt{26}}{2} - \sqrt{26}$

Step 3.2.5.4

Subtract $15$ from $20$ .

$y = \frac{5 + 2 \sqrt{26}}{2} - \sqrt{26}$

Step 3.2.6

To write $- \sqrt{26}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{5 + 2 \sqrt{26}}{2} - \sqrt{26} \cdot \frac{2}{2}$

Step 3.2.7

Combine fractions.

Tap for more steps...

Step 3.2.7.1

Combine $- \sqrt{26}$ and $\frac{2}{2}$ .

$y = \frac{5 + 2 \sqrt{26}}{2} + \frac{- \sqrt{26} \cdot 2}{2}$

Step 3.2.7.2

Combine the numerators over the common denominator.

$y = \frac{5 + 2 \sqrt{26} - \sqrt{26} \cdot 2}{2}$

Step 3.2.8

Simplify the numerator.

Tap for more steps...

Step 3.2.8.1

Multiply $2$ by $- 1$ .

$y = \frac{5 + 2 \sqrt{26} - 2 \sqrt{26}}{2}$

Step 3.2.8.2

Subtract $2 \sqrt{26}$ from $2 \sqrt{26}$ .

$y = \frac{5 + 0}{2}$

Step 3.2.8.3

Add $5$ and $0$ .

$y = \frac{5}{2}$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{2 - \sqrt{26}}{2}, \frac{5}{2})$

$(- \frac{2 + \sqrt{26}}{2}, \frac{5}{2})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(- \frac{2 - \sqrt{26}}{2}, \frac{5}{2}), (- \frac{2 + \sqrt{26}}{2}, \frac{5}{2})$

Equation Form:

$x = - \frac{2 - \sqrt{26}}{2}, y = \frac{5}{2}$

$x = - \frac{2 + \sqrt{26}}{2}, y = \frac{5}{2}$

Step 6