Question

7b. lin uses the quadratic formula: lin knows 36 − 40 is a negative number and isn’t sure what to do next. show how lin can write her solution using i. $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $x = \frac{-(6) \pm \sqrt{(-6)^2 - 4(1)(10)}}{2(1)}$ $x = \frac{6 \pm \sqrt{36 - 40}}{2}$ upload 1 supported file. max 10 mb.

Explanation:

Step1: Simplify the discriminant

First, calculate the value inside the square root: \(36 - 40=- 4\). So the equation becomes \(x=\frac{6\pm\sqrt{- 4}}{2}\).

Step2: Rewrite the square root of negative number

Recall that for a negative number \( - n\) (where \(n>0\)), \(\sqrt{-n}=\sqrt{n}\times\sqrt{-1}\), and by definition, \(\sqrt{-1} = i\). So \(\sqrt{-4}=\sqrt{4}\times\sqrt{-1}\).
Since \(\sqrt{4} = 2\) and \(\sqrt{-1}=i\), we have \(\sqrt{-4}=2i\).

Step3: Substitute back and simplify

Substitute \(\sqrt{-4} = 2i\) into the equation for \(x\): \(x=\frac{6\pm2i}{2}\).
Then, we can factor out a 2 from the numerator: \(x=\frac{2(3\pm i)}{2}\).
Finally, cancel out the 2 in the numerator and the denominator: \(x = 3\pm i\).

Answer:

The solution using \(i\) is \(x = 3 + i\) or \(x=3 - i\) (or combined as \(x=3\pm i\)).