Find the Equation Given the Roots -i square root of 7

$- i \sqrt{7}$

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 $x = - i \sqrt{7}$ was found by solving for $x$ when $x - (- i \sqrt{7}) = y$ and $y = 0$ .

The factor is $x + \sqrt{7} i$

Step 3

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

The factor is $x - i \sqrt{7}$

Step 4

Combine all the factors into a single equation.

$y = (x + \sqrt{7} i) (x - i \sqrt{7})$

Step 5

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

Expand $(x + \sqrt{7} i) (x - i \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$y = x (x - i \sqrt{7}) + \sqrt{7} i (x - i \sqrt{7})$

Apply the distributive property.

$y = x \cdot x + x (- i \sqrt{7}) + \sqrt{7} i (x - i \sqrt{7})$

Apply the distributive property.

$y = x \cdot x + x (- i \sqrt{7}) + \sqrt{7} i x + \sqrt{7} i (- i \sqrt{7})$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- i \sqrt{7}) + \sqrt{7} i x + \sqrt{7} i (- i \sqrt{7})$ .

Reorder the factors in the terms $x (- i \sqrt{7})$ and $\sqrt{7} i x$ .

$y = x \cdot x - i x \sqrt{7} + i x \sqrt{7} + \sqrt{7} i (- i \sqrt{7})$

Add $- i x \sqrt{7}$ and $i x \sqrt{7}$ .

$y = x \cdot x + 0 + \sqrt{7} i (- i \sqrt{7})$

Add $x \cdot x$ and $0$ .

$y = x \cdot x + \sqrt{7} i (- i \sqrt{7})$

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $\sqrt{7} i (- i \sqrt{7})$ .

Raise $\sqrt{7}$ to the power of $1$ .

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

Raise $\sqrt{7}$ to the power of $1$ .

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

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

$y = x^{2} + i (- i {\sqrt{7}}^{1 + 1})$

Add $1$ and $1$ .

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

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

$y = x^{2} - i i {\sqrt{7}}^{2}$

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

$y = x^{2} - i i {\sqrt{7}}^{2}$

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

$y = x^{2} - i^{1 + 1} {\sqrt{7}}^{2}$

Add $1$ and $1$ .

$y = x^{2} - i^{2} {\sqrt{7}}^{2}$

Rewrite $i^{2}$ as $- 1$ .

$y = x^{2} + 1 {\sqrt{7}}^{2}$

Multiply $- 1$ by $- 1$ .

$y = x^{2} + 1 {\sqrt{7}}^{2}$

Multiply ${\sqrt{7}}^{2}$ by $1$ .

$y = x^{2} + {\sqrt{7}}^{2}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$y = x^{2} + 7$

Evaluate the exponent.

$y = x^{2} + 7$

Step 6

Find the Equation Given the Roots -i square root of 7

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