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

Solution:

Step 1:

Identify the roots where the function crosses the x-axis, which occurs at $y=0$.

Step 2:

The root $x=-i\sqrt{7}$ is determined by setting $x-(-i\sqrt{7})=y$ to zero, resulting in the factor $x+i\sqrt{7}$.

Step 3:

Similarly, the root $x=i\sqrt{7}$ is found by setting $x-i\sqrt{7}=y$ to zero, yielding the factor $x-i\sqrt{7}$.

Step 4:

Form the equation by multiplying the factors: $y=(x+i\sqrt{7})(x-i\sqrt{7})$.

Step 5:

Expand the equation by multiplying the factors.

Step 5.1:

Use the FOIL method to expand $(x+i\sqrt{7})(x-i\sqrt{7})$.

Step 5.1.1:

Distribute the terms: $y=x(x-i\sqrt{7})+i\sqrt{7}(x-i\sqrt{7})$.

Step 5.1.2:

Continue distribution: $y=x^2-xi\sqrt{7}+i\sqrt{7}x-i^2(\sqrt{7}{)}^2$.

Step 5.2:

Simplify the expression.

Step 5.2.1:

Combine like terms: $y=x^2-xi\sqrt{7}+xi\sqrt{7}-i^2(\sqrt{7}{)}^2$.

Step 5.2.2:

Simplify each term.

Step 5.2.2.1:

Calculate $x^2$: $y=x^2-i^2(\sqrt{7}{)}^2$.

Step 5.2.2.2:

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

Step 5.2.2.3:

Simplify the square of $\sqrt{7}$: $y=x^2+7$.

Step 6:

The final equation is $y=x^2+7$.

Knowledge Notes:

1. Roots of a Polynomial: The roots of a polynomial are the values of $x$ for which the polynomial equals zero. In the context of a quadratic equation, these are the points where the graph of the equation crosses the x-axis.

2. Complex Conjugates: If a polynomial has real coefficients and a complex root, the complex conjugate of that root is also a root. In this case, the roots are $-i\sqrt{7}$ and $i\sqrt{7}$, which are complex conjugates.

3. Multiplying Complex Numbers: When multiplying complex numbers, one should use the distributive property, also known as the FOIL method for binomials, to expand the product. The product of a complex number and its conjugate results in a real number.

4. Imaginary Unit: The imaginary unit $i$ is defined such that $i^2=-1$. This property is used to simplify expressions involving $i$.

5. Squaring a Square Root: When a square root is squared, such as $(\sqrt{7}{)}^2$, it simplifies to the number under the root, which is 7 in this case.

6. Simplifying Expressions: The process of simplifying expressions involves combining like terms, applying the distributive property, and reducing expressions to their simplest form.

7. Quadratic Equations: A quadratic equation in standard form is written as $y=a{x}^2+bx+c$. When given the roots, the equation can be formed by multiplying the factors associated with each root.