Answer:
The first one
Step-by-step explanation:
We want to find the roots of the equation -2x + 3 = -8x²
Step 1: Our first step is to get the equation in quadratic form so we can use the quadratic formula to find the roots
Quadratic form: ax² + bx + c = 0
We can easily get this equation in quadratic form by moving -8x² to the right side. We can do this using inverse operations. The inverse of subtraction is addition so to get rid of -8x² we add 8x² to both sides
After adding -8x² to both sides we acquire 8x² - 2x + 3 = 0
The equation is now in quadratic form meaning we can now use the quadratic formula to find the roots.
Quadratic Formula : [tex]\frac{-b+-\sqrt{b^2-4(a)(c)} }{2(a)}[/tex]
where the values of a, b and c are derived from the equation
Remember that the equation is in quadratic form ax² + bx + c = 0
8x² - 2x + 3 = 0 so a = 8 , b = - 2 and c = 3
We then plug in these values into the quadratic formula ( note that the +- means plus or minus meaning that we have to evaluate this twice, once when the discriminant ( b² - 4(a)(c) is the discriminant ) is being add to -b and once when the discriminant is being subtracted from -b )
First lets evaluate when the discriminant is being added to -b
Recall the quadratic formula : [tex]\frac{-b+\sqrt{b^2-4(a)(c)} }{2(a)}[/tex]
a = 8 , b = - 2 and c = 3
[tex]\frac{2+\sqrt{2^2-4(8)(3)} }{2(8)}[/tex]
Work being done inside of the square root: 2² = 4 , -4 * 8 = -32 , -32 * 3 = -96
4 - 96 = - 92
Work being done at denominator : 2 * 8 = 16
[tex]\frac{2+\sqrt{-92} }{16}[/tex]
The first root is [tex]\frac{2+\sqrt{-92} }{16}[/tex]
We now do this same process but instead we subtract the discriminant.
We would be left with the same thing but it would be [tex]\frac{2-\sqrt{-92} }{16}[/tex] instead of [tex]\frac{2+\sqrt{-92} }{16}[/tex]
In some cases we would get a completely different answer, so evaluating it twice, once when the discriminant is being add to -b and once when the discriminant is being subtracted from -b may be important in some cases.
We then simplify the two roots. You may notice that there is a negative number under the radical and you might ask how can you square root a negative? well you can't which is when imaginary roots come in. Imaginary roots: i = -1 . We can take out an i from -92 making it 92 because i = -1 and -92/-1 = 92. We would be left with i√92
So we can conclude that the roots of the equation are [tex]\frac{2+i\sqrt{92} }{16} and \frac{2-i\sqrt{92} }{16}[/tex]
Looking at the answer choices we notice that there are two very similar answers. The first and second one. The only difference between the two is that 2 is positive on the first one and 2 is negative on the second one. Looking at the roots we just found, the 2 should be positive therefore the answer is the first one.
Note that ± means plus or minus and it means that the expression can either be added or subtracted and it will be a root. This means that saying the roots are [tex]\frac{2+-i\sqrt{92} }{16}[/tex] is the same as saying the roots are [tex]\frac{2+i\sqrt{92} }{16} and \frac{2-i\sqrt{92} }{16}[/tex]