Question

sing the quadratic formula with complex solutions
open - ended question
solutions are imaginary numbers/complex numbers
$x = 2 + i\sqrt{3}$ and $x = 2 - i\sqrt{3}$
type your answer here

Explanation:

Step1: Rewrite to standard quadratic form

Start with $x^2 + 3x + 7 = 7x$, subtract $7x$ from both sides:
$x^2 - 4x + 7 = 0$

Step2: Identify coefficients

For $ax^2+bx+c=0$, we get:
$a=1$, $b=-4$, $c=7$

Step3: Substitute into quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$x=\frac{-(-4)\pm\sqrt{(-4)^2-4(1)(7)}}{2(1)}$

Step4: Simplify discriminant and numerator

Calculate discriminant and simplify:
$x=\frac{4\pm\sqrt{16-28}}{2}=\frac{4\pm\sqrt{-12}}{2}$

Step5: Rewrite root of negative number

Express $\sqrt{-12}$ as $i\sqrt{12}=2i\sqrt{3}$:
$x=\frac{4\pm2i\sqrt{3}}{2}$

Step6: Simplify the fraction

Divide each term in numerator by 2:
$x=2\pm i\sqrt{3}$

Answer:

$x=2+i\sqrt{3}$ and $x=2-i\sqrt{3}$