Question

PLEASE HELP! ♥ Solve the system of equations.
y = 2x² – 3
y = 3x – 1

Answer 1

for what? i mean slope or intersection or what

Answer 2

not sure, it doesnt say

Answer 3

You can do this: 2x² - 3 = 3x - 1, because both are equal to y.
Now it is a quadratic equation.
If you write it as 2x² - 3x - 2 = 0, you can use the Quadratic Formula to solve it for x.

Answer 4

Or, maybe you can solve it by factoring.

Answer 5

a. no solution
b. (-1/r2, 5), (2, –5/2)
c. (–1/2, –5/2), (2, 5)
d. (1/2, 5/2), (2, 5)

Answer 6

Do you know the Quadratic Formula?

Answer 7

no i dont

Answer 8

It is the formula that gives you the solutions of any quadratic equation:

If \(ax^2+bx+c=0\), then \(x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\).

So all you have to do is: look at the values of a, b and c in your equation:

2x² - 3x - 2 = 0, so

a = 2
b = -3
c = -2

Plug these values in the Quadractic formula and out come your solutions!
Once you've the solutions for x, you can substitute them in one of your original equations (y=3x-1 is the simplest), to calculate y.
THen you have two pairs (x, y).

Compare them with the answer options to see which one to choose.

Answer 9

If you don't know the Quadratic Formula, chances are that up until now you always have solved quadratic equations by factoring.
This is a good technique, although it only works if the solutions are "nice" numbers, like integers or fractions like ½, ¼ and so on.

Looking at the y-values of the answer options, this seems the case.

If we want to solve 2x² - 3x - 2 = 0 by factoring, we must write this equation as:

(a + b)(c + d), which, aftwer FOILing (calculating all possible multiplications of a and b with c and d) should give 2x² - 3x - 2.

This is not so hard as it seems, because although we have to come up with a, b, c and d, most of these numbers are already given!

FOIL means: First, Outer, Inner, Last, so

(a + b)(c + d)=
(First) ac +
(Outer) ad +
(Inner) bc +
(Last) bd

all in all, this gives ac+ad+bc+bd.

Now if we are to get 2x² - 3x - 2, we have to write it like this:

(2x ...)(x ...), because only then 2x times x is 2x² (=the First of FOIL).
Also, we don't have so much to choose for the Last part, because it means that the two numbers that go on the dots have product -2.

If these numbers are "nice" as I claimed earlier, then they sould be something like 1 and -2 (product is -2), or -1 and 2 (also product -2), or even 4 and -½.

Now what is it?

Look at (2x+1)(x-2) = 2x² -4x+1x-2, so the middle two terms are -3x when added.

So we have found the numbers we were looking for!

2x² - 3x - 2 = 0 is thus the same as (2x + 1)(x - 2) = 0.

You should be able to spot the solutions for x now...