Find the Equation Given the Roots (-7+ square root of 13)/2 , (-7- square root of 13)/2

$\frac{- 7 + \sqrt{13}}{2}$ , $\frac{- 7 - \sqrt{13}}{2}$

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

$y = 0$ at the roots

The root at $x = - \frac{7 - \sqrt{13}}{2}$ was found by solving for $x$ when $x - (- \frac{7 - \sqrt{13}}{2}) = y$ and $y = 0$ .

The factor is $\frac{2 x + 7 - \sqrt{13}}{2}$

The root at $x = - \frac{7 + \sqrt{13}}{2}$ was found by solving for $x$ when $x - (- \frac{7 + \sqrt{13}}{2}) = y$ and $y = 0$ .

The factor is $\frac{2 x + 7 + \sqrt{13}}{2}$

Combine all the factors into a single equation.

$y = (\frac{2 x + 7 - \sqrt{13}}{2}) (\frac{2 x + 7 + \sqrt{13}}{2})$

Multiply all the factors to simplify the equation $y = (\frac{2 x + 7 - \sqrt{13}}{2}) (\frac{2 x + 7 + \sqrt{13}}{2})$ .

Multiply $(\frac{2 x + 7 - \sqrt{13}}{2}) (\frac{2 x + 7 + \sqrt{13}}{2})$ .

Multiply $\frac{2 x + 7 - \sqrt{13}}{2}$ by $\frac{2 x + 7 + \sqrt{13}}{2}$ .

$y = \frac{(2 x + 7 - \sqrt{13}) (2 x + 7 + \sqrt{13})}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{(2 x + 7 - \sqrt{13}) (2 x + 7 + \sqrt{13})}{4}$

Expand $(2 x + 7 - \sqrt{13}) (2 x + 7 + \sqrt{13})$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{2 x (2 x) + 2 x \cdot 7 + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{2 \cdot (2 x \cdot x) + 2 x \cdot 7 + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

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

Move $x$ .

$y = \frac{2 \cdot (2 (x \cdot x)) + 2 x \cdot 7 + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $x$ by $x$ .

$y = \frac{2 \cdot (2 x^{2}) + 2 x \cdot 7 + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $2$ by $2$ .

$y = \frac{4 x^{2} + 2 x \cdot 7 + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $7$ by $2$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 7 (2 x) + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $2$ by $7$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 7 \cdot 7 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $7$ by $7$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - \sqrt{13} (2 x) - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $2$ by $- 1$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - \sqrt{13} \cdot 7 - \sqrt{13} \sqrt{13}}{4}$

Multiply $7$ by $- 1$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - \sqrt{13} \sqrt{13}}{4}$

Multiply $- \sqrt{13} \sqrt{13}$ .

Raise $\sqrt{13}$ to the power of $1$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - (\sqrt{13} \sqrt{13})}{4}$

Raise $\sqrt{13}$ to the power of $1$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - (\sqrt{13} \sqrt{13})}{4}$

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

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - {\sqrt{13}}^{1 + 1}}{4}$

Add $1$ and $1$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - {\sqrt{13}}^{2}}{4}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - {(13^{\frac{1}{2}})}^{2}}{4}$

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

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13^{\frac{1}{2} \cdot 2}}{4}$

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

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13^{\frac{2}{2}}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13^{\frac{2}{2}}}{4}$

Rewrite the expression.

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13}{4}$

Evaluate the exponent.

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 1 \cdot 13}{4}$

Multiply $- 1$ by $13$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13}{4}$

Combine the opposite terms in $4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 \sqrt{13} x - 7 \sqrt{13} - 13$ .

Reorder the factors in the terms $2 x \sqrt{13}$ and $- 2 \sqrt{13} x$ .

$y = \frac{4 x^{2} + 14 x + 2 x \sqrt{13} + 14 x + 49 + 7 \sqrt{13} - 2 x \sqrt{13} - 7 \sqrt{13} - 13}{4}$

Subtract $2 x \sqrt{13}$ from $2 x \sqrt{13}$ .

$y = \frac{4 x^{2} + 14 x + 0 + 14 x + 49 + 7 \sqrt{13} - 7 \sqrt{13} - 13}{4}$

Add $4 x^{2} + 14 x$ and $0$ .

$y = \frac{4 x^{2} + 14 x + 14 x + 49 + 7 \sqrt{13} - 7 \sqrt{13} - 13}{4}$

Subtract $7 \sqrt{13}$ from $7 \sqrt{13}$ .

$y = \frac{4 x^{2} + 14 x + 14 x + 49 + 0 - 13}{4}$

Add $4 x^{2} + 14 x + 14 x + 49$ and $0$ .

$y = \frac{4 x^{2} + 14 x + 14 x + 49 - 13}{4}$

Add $14 x$ and $14 x$ .

$y = \frac{4 x^{2} + 28 x + 49 - 13}{4}$

Subtract $13$ from $49$ .

$y = \frac{4 x^{2} + 28 x + 36}{4}$

Split the fraction $\frac{4 x^{2} + 28 x + 36}{4}$ into two fractions.

$y = \frac{4 x^{2} + 28 x}{4} + \frac{36}{4}$

Split the fraction $\frac{4 x^{2} + 28 x}{4}$ into two fractions.

$y = \frac{4 x^{2}}{4} + \frac{28 x}{4} + \frac{36}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$y = \frac{4 x^{2}}{4} + \frac{28 x}{4} + \frac{36}{4}$

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

$y = x^{2} + \frac{28 x}{4} + \frac{36}{4}$

Cancel the common factor of $28$ and $4$ .

Factor $4$ out of $28 x$ .

$y = x^{2} + \frac{4 (7 x)}{4} + \frac{36}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$y = x^{2} + \frac{4 (7 x)}{4 (1)} + \frac{36}{4}$

Cancel the common factor.

$y = x^{2} + \frac{4 (7 x)}{4 \cdot 1} + \frac{36}{4}$

Rewrite the expression.

$y = x^{2} + \frac{7 x}{1} + \frac{36}{4}$

Divide $7 x$ by $1$ .

$y = x^{2} + 7 x + \frac{36}{4}$

Divide $36$ by $4$ .

$y = x^{2} + 7 x + 9$

Find the Equation Given the Roots (-7+ square root of 13)/2 , (-7- square root of 13)/2

Related Questions

Support LCMGCF.com by disabling your adblocker.