Algebra Examples

$y = 3 x + 5$ $y = 4 x^{2} + x$

Step 1

Eliminate the equal sides of each equation and combine.

$3 x + 5 = 4 x^{2} + x$

Step 2

Solve $3 x + 5 = 4 x^{2} + x$ for $x$ .

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Step 2.1

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

$4 x^{2} + x = 3 x + 5$

Step 2.2

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

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Step 2.2.1

Subtract $3 x$ from both sides of the equation.

$4 x^{2} + x - 3 x = 5$

Step 2.2.2

Subtract $3 x$ from $x$ .

$4 x^{2} - 2 x = 5$

Step 2.3

Subtract $5$ from both sides of the equation.

$4 x^{2} - 2 x - 5 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 4$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (4 \cdot - 5)}}{2 \cdot 4} + y = 4 x^{2} + x$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Step 2.6.1.2

Multiply $- 4 \cdot 4 \cdot - 5$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Step 2.6.1.2.2

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Step 2.6.1.3

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Step 2.6.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.6.1.4.1

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Step 2.6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Step 2.6.2

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Step 2.6.3

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Step 2.7.1.2

Multiply $- 4 \cdot 4 \cdot - 5$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Step 2.7.1.2.2

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Step 2.7.1.3

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Step 2.7.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.7.1.4.1

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Step 2.7.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Step 2.7.2

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Step 2.7.3

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{21}}{4}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Step 2.8.1.2

Multiply $- 4 \cdot 4 \cdot - 5$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Step 2.8.1.2.2

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Step 2.8.1.3

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Step 2.8.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.8.1.4.1

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Step 2.8.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Step 2.8.2

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Step 2.8.3

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{21}}{4}$

Step 2.9

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{21}}{4}, \frac{1 - \sqrt{21}}{4}$

Step 3

Evaluate $y$ when $x = \frac{1 + \sqrt{21}}{4}$ .

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Step 3.1

Substitute $\frac{1 + \sqrt{21}}{4}$ for $x$ .

$y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2

Substitute $\frac{1 + \sqrt{21}}{4}$ for $x$ in $y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2

Simplify $4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Apply the product rule to $\frac{1 + \sqrt{21}}{4}$ .

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4^{2}} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.2

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

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{16} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.3

Cancel the common factor of $4$ .

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Step 3.2.2.1.3.1

Factor $4$ out of $16$ .

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4 (4)} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.3.2

Cancel the common factor.

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4 \cdot 4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.3.3

Rewrite the expression.

$y = \frac{{(1 + \sqrt{21})}^{2}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.4

Rewrite ${(1 + \sqrt{21})}^{2}$ as $(1 + \sqrt{21}) (1 + \sqrt{21})$ .

$y = \frac{(1 + \sqrt{21}) (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.5

Expand $(1 + \sqrt{21}) (1 + \sqrt{21})$ using the FOIL Method.

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Step 3.2.2.1.5.1

Apply the distributive property.

$y = \frac{1 (1 + \sqrt{21}) + \sqrt{21} (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.5.2

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{21} + \sqrt{21} (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.5.3

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6

Simplify and combine like terms.

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Step 3.2.2.1.6.1

Simplify each term.

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Step 3.2.2.1.6.1.1

Multiply $1$ by $1$ .

$y = \frac{1 + 1 \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.2

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.3

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.4

Combine using the product rule for radicals.

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21 \cdot 21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.5

Multiply $21$ by $21$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{441}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.6

Rewrite $441$ as $21^{2}$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21^{2}}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.1.7

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

$y = \frac{1 + \sqrt{21} + \sqrt{21} + 21}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.2

Add $1$ and $21$ .

$y = \frac{22 + \sqrt{21} + \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.6.3

Add $\sqrt{21}$ and $\sqrt{21}$ .

$y = \frac{22 + 2 \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7

Cancel the common factor of $22 + 2 \sqrt{21}$ and $4$ .

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Step 3.2.2.1.7.1

Factor $2$ out of $22$ .

$y = \frac{2 \cdot 11 + 2 \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7.2

Factor $2$ out of $2 \sqrt{21}$ .

$y = \frac{2 \cdot 11 + 2 (\sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7.3

Factor $2$ out of $2 \cdot 11 + 2 (\sqrt{21})$ .

$y = \frac{2 \cdot (11 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7.4

Cancel the common factors.

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Step 3.2.2.1.7.4.1

Factor $2$ out of $4$ .

$y = \frac{2 \cdot (11 + \sqrt{21})}{2 (2)} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7.4.2

Cancel the common factor.

$y = \frac{2 \cdot (11 + \sqrt{21})}{2 \cdot 2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.1.7.4.3

Rewrite the expression.

$y = \frac{11 + \sqrt{21}}{2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.2

To write $\frac{11 + \sqrt{21}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{11 + \sqrt{21}}{2} \cdot \frac{2}{2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.2.3.1

Multiply $\frac{11 + \sqrt{21}}{2}$ by $\frac{2}{2}$ .

$y = \frac{(11 + \sqrt{21}) \cdot 2}{2 \cdot 2} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.3.2

Multiply $2$ by $2$ .

$y = \frac{(11 + \sqrt{21}) \cdot 2}{4} + \frac{1 + \sqrt{21}}{4}$

Step 3.2.2.4

Combine the numerators over the common denominator.

$y = \frac{(11 + \sqrt{21}) \cdot 2 + 1 + \sqrt{21}}{4}$

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Apply the distributive property.

$y = \frac{11 \cdot 2 + \sqrt{21} \cdot 2 + 1 + \sqrt{21}}{4}$

Step 3.2.2.5.2

Multiply $11$ by $2$ .

$y = \frac{22 + \sqrt{21} \cdot 2 + 1 + \sqrt{21}}{4}$

Step 3.2.2.5.3

Move $2$ to the left of $\sqrt{21}$ .

$y = \frac{22 + 2 \cdot \sqrt{21} + 1 + \sqrt{21}}{4}$

Step 3.2.2.5.4

Add $22$ and $1$ .

$y = \frac{23 + 2 \sqrt{21} + \sqrt{21}}{4}$

Step 3.2.2.5.5

Add $2 \sqrt{21}$ and $\sqrt{21}$ .

$y = \frac{23 + 3 \sqrt{21}}{4}$

Step 4

Evaluate $y$ when $x = \frac{1 - \sqrt{21}}{4}$ .

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Step 4.1

Substitute $\frac{1 - \sqrt{21}}{4}$ for $x$ .

$y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2

Substitute $\frac{1 - \sqrt{21}}{4}$ for $x$ in $y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2

Simplify $4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$ .

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Apply the product rule to $\frac{1 - \sqrt{21}}{4}$ .

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4^{2}} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.2

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

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{16} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.3

Cancel the common factor of $4$ .

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Step 4.2.2.1.3.1

Factor $4$ out of $16$ .

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4 (4)} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.3.2

Cancel the common factor.

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4 \cdot 4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.3.3

Rewrite the expression.

$y = \frac{{(1 - \sqrt{21})}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.4

Rewrite ${(1 - \sqrt{21})}^{2}$ as $(1 - \sqrt{21}) (1 - \sqrt{21})$ .

$y = \frac{(1 - \sqrt{21}) (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.5

Expand $(1 - \sqrt{21}) (1 - \sqrt{21})$ using the FOIL Method.

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Step 4.2.2.1.5.1

Apply the distributive property.

$y = \frac{1 (1 - \sqrt{21}) - \sqrt{21} (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.5.2

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{21}) - \sqrt{21} (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.5.3

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{21}) - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6

Simplify and combine like terms.

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Step 4.2.2.1.6.1

Simplify each term.

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Step 4.2.2.1.6.1.1

Multiply $1$ by $1$ .

$y = \frac{1 + 1 (- \sqrt{21}) - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.2

Multiply $- \sqrt{21}$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.3

Multiply $- 1$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4

Multiply $- \sqrt{21} (- \sqrt{21})$ .

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Step 4.2.2.1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 1 \sqrt{21} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4.2

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + \sqrt{21} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4.3

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4.4

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1} {\sqrt{21}}^{1}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4.5

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

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1 + 1}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.4.6

Add $1$ and $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 4.2.2.1.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {(21^{\frac{1}{2}})}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5.2

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

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{1}{2} \cdot 2}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5.3

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

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{2}{2}}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.6.1.5.4.1

Cancel the common factor.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{2}{2}}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5.4.2

Rewrite the expression.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{1}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.1.5.5

Evaluate the exponent.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.2

Add $1$ and $21$ .

$y = \frac{22 - \sqrt{21} - \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.6.3

Subtract $\sqrt{21}$ from $- \sqrt{21}$ .

$y = \frac{22 - 2 \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7

Cancel the common factor of $22 - 2 \sqrt{21}$ and $4$ .

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Step 4.2.2.1.7.1

Factor $2$ out of $22$ .

$y = \frac{2 (11) - 2 \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7.2

Factor $2$ out of $- 2 \sqrt{21}$ .

$y = \frac{2 (11) + 2 (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7.3

Factor $2$ out of $2 (11) + 2 (- \sqrt{21})$ .

$y = \frac{2 (11 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7.4

Cancel the common factors.

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Step 4.2.2.1.7.4.1

Factor $2$ out of $4$ .

$y = \frac{2 (11 - \sqrt{21})}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7.4.2

Cancel the common factor.

$y = \frac{2 (11 - \sqrt{21})}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.1.7.4.3

Rewrite the expression.

$y = \frac{11 - \sqrt{21}}{2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.2

To write $\frac{11 - \sqrt{21}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{11 - \sqrt{21}}{2} \cdot \frac{2}{2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.3.1

Multiply $\frac{11 - \sqrt{21}}{2}$ by $\frac{2}{2}$ .

$y = \frac{(11 - \sqrt{21}) \cdot 2}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.3.2

Multiply $2$ by $2$ .

$y = \frac{(11 - \sqrt{21}) \cdot 2}{4} + \frac{1 - \sqrt{21}}{4}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$y = \frac{(11 - \sqrt{21}) \cdot 2 + 1 - \sqrt{21}}{4}$

Step 4.2.2.5

Simplify the numerator.

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Step 4.2.2.5.1

Apply the distributive property.

$y = \frac{11 \cdot 2 - \sqrt{21} \cdot 2 + 1 - \sqrt{21}}{4}$

Step 4.2.2.5.2

Multiply $11$ by $2$ .

$y = \frac{22 - \sqrt{21} \cdot 2 + 1 - \sqrt{21}}{4}$

Step 4.2.2.5.3

Multiply $2$ by $- 1$ .

$y = \frac{22 - 2 \sqrt{21} + 1 - \sqrt{21}}{4}$

Step 4.2.2.5.4

Add $22$ and $1$ .

$y = \frac{23 - 2 \sqrt{21} - \sqrt{21}}{4}$

Step 4.2.2.5.5

Subtract $\sqrt{21}$ from $- 2 \sqrt{21}$ .

$y = \frac{23 - 3 \sqrt{21}}{4}$

Step 5

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

$(\frac{1 + \sqrt{21}}{4}, \frac{23 + 3 \sqrt{21}}{4})$

$(\frac{1 - \sqrt{21}}{4}, \frac{23 - 3 \sqrt{21}}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1 + \sqrt{21}}{4}, \frac{23 + 3 \sqrt{21}}{4}), (\frac{1 - \sqrt{21}}{4}, \frac{23 - 3 \sqrt{21}}{4})$

Equation Form:

$x = \frac{1 + \sqrt{21}}{4}, y = \frac{23 + 3 \sqrt{21}}{4}$

$x = \frac{1 - \sqrt{21}}{4}, y = \frac{23 - 3 \sqrt{21}}{4}$

Step 7