Math  /  Algebra

Question6. 5x2+4=765 x^{2}+4=-76 \begin{tabular}{|l|l|} \hline Method: & Why? \\ \hline \end{tabular}
Solve:

Studdy Solution

STEP 1

What is this asking? We need to find the value(s) of xx that make the equation 5x2+4=765x^2 + 4 = -76 true. Watch out! Remember that squaring a number always results in a positive value (or zero).
This might be important later!

STEP 2

1. Isolate the x2x^2 term
2. Solve for xx

STEP 3

We want to get xx by itself, so let's start by moving the **constant term**, 4, to the other side of the equation.
To do this, we'll subtract 4 from both sides of the equation.
Remember, what we do to one side, we *must* do to the other to keep the equation balanced!
5x2+44=7645x^2 + 4 - 4 = -76 - 45x2=805x^2 = -80

STEP 4

Now, we want to isolate x2x^2.
Since x2x^2 is being multiplied by **5**, we'll **divide** both sides of the equation by **5** to get x2x^2 all alone.
5x25=805\frac{5x^2}{5} = \frac{-80}{5}x2=16x^2 = -16

STEP 5

To solve for xx, we need to undo the squaring.
We do this by taking the square root of both sides.
But hold on!
Remember what we said earlier?
Squaring a real number *always* gives a positive result (or zero).
So, can x2x^2 ever equal a negative number like **-16** if xx is a real number?
Nope!
x2=16\sqrt{x^2} = \sqrt{-16}

STEP 6

Since we can't take the square root of a negative number in the real number system, we need to introduce the **imaginary unit**, denoted by ii, where i2=1i^2 = -1.
This allows us to rewrite 16\sqrt{-16} as 161=161=4i\sqrt{16 \cdot -1} = \sqrt{16} \cdot \sqrt{-1} = 4i.
x=±4ix = \pm 4i

STEP 7

The solutions to the equation 5x2+4=765x^2 + 4 = -76 are x=4ix = 4i and x=4ix = -4i.

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