Algebra Examples

$\frac{- 9 + \sqrt{57}}{4}$ , $\frac{- 9 - \sqrt{57}}{4}$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - \frac{9 - \sqrt{57}}{4}$ was found by solving for $x$ when $x - (- \frac{9 - \sqrt{57}}{4}) = y$ and $y = 0$ .

The factor is $\frac{4 x + 9 - \sqrt{57}}{4}$

Step 3

The root at $x = - \frac{9 + \sqrt{57}}{4}$ was found by solving for $x$ when $x - (- \frac{9 + \sqrt{57}}{4}) = y$ and $y = 0$ .

The factor is $\frac{4 x + 9 + \sqrt{57}}{4}$

Step 4

Combine all the factors into a single equation.

$y = (\frac{4 x + 9 - \sqrt{57}}{4}) (\frac{4 x + 9 + \sqrt{57}}{4})$

Step 5

Multiply all the factors to simplify the equation $y = (\frac{4 x + 9 - \sqrt{57}}{4}) (\frac{4 x + 9 + \sqrt{57}}{4})$ .

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Step 5.1

Multiply $(\frac{4 x + 9 - \sqrt{57}}{4}) (\frac{4 x + 9 + \sqrt{57}}{4})$ .

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Step 5.1.1

Multiply $\frac{4 x + 9 - \sqrt{57}}{4}$ by $\frac{4 x + 9 + \sqrt{57}}{4}$ .

$y = \frac{(4 x + 9 - \sqrt{57}) (4 x + 9 + \sqrt{57})}{4 \cdot 4}$

Step 5.1.2

Multiply $4$ by $4$ .

$y = \frac{(4 x + 9 - \sqrt{57}) (4 x + 9 + \sqrt{57})}{16}$

Step 5.2

Expand $(4 x + 9 - \sqrt{57}) (4 x + 9 + \sqrt{57})$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{4 x (4 x) + 4 x \cdot 9 + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3

Simplify each term.

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Step 5.3.1

Rewrite using the commutative property of multiplication.

$y = \frac{4 \cdot (4 x \cdot x) + 4 x \cdot 9 + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.2.1

Move $x$ .

$y = \frac{4 \cdot (4 (x \cdot x)) + 4 x \cdot 9 + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.2.2

Multiply $x$ by $x$ .

$y = \frac{4 \cdot (4 x^{2}) + 4 x \cdot 9 + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.3

Multiply $4$ by $4$ .

$y = \frac{16 x^{2} + 4 x \cdot 9 + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.4

Multiply $9$ by $4$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 9 (4 x) + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.5

Multiply $4$ by $9$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 9 \cdot 9 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.6

Multiply $9$ by $9$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - \sqrt{57} (4 x) - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.7

Multiply $4$ by $- 1$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - \sqrt{57} \cdot 9 - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.8

Multiply $9$ by $- 1$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - \sqrt{57} \sqrt{57}}{16}$

Step 5.3.9

Multiply $- \sqrt{57} \sqrt{57}$ .

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Step 5.3.9.1

Raise $\sqrt{57}$ to the power of $1$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - (\sqrt{57} \sqrt{57})}{16}$

Step 5.3.9.2

Raise $\sqrt{57}$ to the power of $1$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - (\sqrt{57} \sqrt{57})}{16}$

Step 5.3.9.3

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

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - {\sqrt{57}}^{1 + 1}}{16}$

Step 5.3.9.4

Add $1$ and $1$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - {\sqrt{57}}^{2}}{16}$

Step 5.3.10

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 5.3.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - {(57^{\frac{1}{2}})}^{2}}{16}$

Step 5.3.10.2

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

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57^{\frac{1}{2} \cdot 2}}{16}$

Step 5.3.10.3

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

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57^{\frac{2}{2}}}{16}$

Step 5.3.10.4

Cancel the common factor of $2$ .

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Step 5.3.10.4.1

Cancel the common factor.

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57^{\frac{2}{2}}}{16}$

Step 5.3.10.4.2

Rewrite the expression.

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57}{16}$

Step 5.3.10.5

Evaluate the exponent.

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 1 \cdot 57}{16}$

Step 5.3.11

Multiply $- 1$ by $57$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57}{16}$

Step 5.4

Combine the opposite terms in $16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 \sqrt{57} x - 9 \sqrt{57} - 57$ .

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Step 5.4.1

Reorder the factors in the terms $4 x \sqrt{57}$ and $- 4 \sqrt{57} x$ .

$y = \frac{16 x^{2} + 36 x + 4 x \sqrt{57} + 36 x + 81 + 9 \sqrt{57} - 4 x \sqrt{57} - 9 \sqrt{57} - 57}{16}$

Step 5.4.2

Subtract $4 x \sqrt{57}$ from $4 x \sqrt{57}$ .

$y = \frac{16 x^{2} + 36 x + 0 + 36 x + 81 + 9 \sqrt{57} - 9 \sqrt{57} - 57}{16}$

Step 5.4.3

Add $16 x^{2} + 36 x$ and $0$ .

$y = \frac{16 x^{2} + 36 x + 36 x + 81 + 9 \sqrt{57} - 9 \sqrt{57} - 57}{16}$

Step 5.4.4

Subtract $9 \sqrt{57}$ from $9 \sqrt{57}$ .

$y = \frac{16 x^{2} + 36 x + 36 x + 81 + 0 - 57}{16}$

Step 5.4.5

Add $16 x^{2} + 36 x + 36 x + 81$ and $0$ .

$y = \frac{16 x^{2} + 36 x + 36 x + 81 - 57}{16}$

Step 5.5

Add $36 x$ and $36 x$ .

$y = \frac{16 x^{2} + 72 x + 81 - 57}{16}$

Step 5.6

Subtract $57$ from $81$ .

$y = \frac{16 x^{2} + 72 x + 24}{16}$

Step 5.7

Split the fraction $\frac{16 x^{2} + 72 x + 24}{16}$ into two fractions.

$y = \frac{16 x^{2} + 72 x}{16} + \frac{24}{16}$

Step 5.8

Split the fraction $\frac{16 x^{2} + 72 x}{16}$ into two fractions.

$y = \frac{16 x^{2}}{16} + \frac{72 x}{16} + \frac{24}{16}$

Step 5.9

Cancel the common factor of $16$ .

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Step 5.9.1

Cancel the common factor.

$y = \frac{16 x^{2}}{16} + \frac{72 x}{16} + \frac{24}{16}$

Step 5.9.2

Divide $x^{2}$ by $1$ .

$y = x^{2} + \frac{72 x}{16} + \frac{24}{16}$

Step 5.10

Cancel the common factor of $72$ and $16$ .

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Step 5.10.1

Factor $8$ out of $72 x$ .

$y = x^{2} + \frac{8 (9 x)}{16} + \frac{24}{16}$

Step 5.10.2

Cancel the common factors.

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Step 5.10.2.1

Factor $8$ out of $16$ .

$y = x^{2} + \frac{8 (9 x)}{8 (2)} + \frac{24}{16}$

Step 5.10.2.2

Cancel the common factor.

$y = x^{2} + \frac{8 (9 x)}{8 \cdot 2} + \frac{24}{16}$

Step 5.10.2.3

Rewrite the expression.

$y = x^{2} + \frac{9 x}{2} + \frac{24}{16}$

Step 5.11

Cancel the common factor of $24$ and $16$ .

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Step 5.11.1

Factor $8$ out of $24$ .

$y = x^{2} + \frac{9 x}{2} + \frac{8 (3)}{16}$

Step 5.11.2

Cancel the common factors.

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Step 5.11.2.1

Factor $8$ out of $16$ .

$y = x^{2} + \frac{9 x}{2} + \frac{8 \cdot 3}{8 \cdot 2}$

Step 5.11.2.2

Cancel the common factor.

$y = x^{2} + \frac{9 x}{2} + \frac{8 \cdot 3}{8 \cdot 2}$

Step 5.11.2.3

Rewrite the expression.

$y = x^{2} + \frac{9 x}{2} + \frac{3}{2}$

Step 6