Maths experts. i need your help pls. Pls show workings and steps.?

Question

By Completing the square

1) X^2- X- 462=

2) the coordinates of 5 points, including the vertex and intercept. of y= -2x^2 + 3x+1

Answer 1

1)

I'm not sure what you want your first equation to equal. I'm going to assume that it's equal to y.

y = x^2 - x - 462

Your first step is to determine "half squared" of the coefficient of x. The coefficient of x is -1, so our key number lies in getting half squared of this. 1/2 of -1 is -1/2, squared is +1/4. So what we do is add 1/4, and, to make sure it's still an equation, subtract 1/4 to offset it.

y = x^2 - x + 1/4 - 462 - 1/4

Now that you added the "half squared" value in, you can group the first three terms into a square.

y = (x - 1/2)^2 - 462 - 1/4

Let's change 462 into a fraction over 4.

y = (x - 1/2)^2 - 1848/4 - 1/4
y = (x - 1/2)^2 - 1849/4

And the square has been completed.

2)
y = -2x^2 + 3x + 1

Your first step is to factor -2 out of the first two terms.

y = -2 (x^2 - (3/2)x) + 1

Now, we need to determine the "half squared" of the coefficient of x, which is 3/2. 1/2 of 3/2 is 3/4 (we just multiply to figure that out), and then 3/4 squared is 9/16. Our key number this time is 9/16, so we add this in.

y = -2 (x^2 - (3/2)x + 9/16) + 1 + ?

Now, last time, offsetting our "half square" value was as simple as subtracting it. This time it's not that simple. We have to determine what goes in the question mark. Remember that since we added 9/16 *within* the brackets, what we really did was add (-2)(9/16) to the equation, due to the -2 outside of the brackets. Therefore, we must offset this by ADDING (2)(9/16).

y = -2 (x^2 - (3/2)x + 9/16) + 1 + 2(9/16)

Now we can express it as a square.

y = -2 (x - (3/4))^2 + 1 + 18/16

y = -2 (x - (3/4))^2 + 16/16 + 18/16
y = -2 (x - (3/4))^2 + 34/16
y = -2 (x - (3/4))^2 + 17/8

For the equation

y = A(x - h)^2 + k

The coordinates of the vertex is given by (h,k). So our vertex in this case is (3/4, 17/8). Note that I took the negative of what's in the brackets to get 3/4.

To obtain the y-intercept, we make x = 0. y = 0 + 0 + 1 = 1
So the y-intercept is 1, and another coordinate would be (0,1)

To obtain the x-intercept, we make y = 0. Then

0 = -2 (x - (3/4))^2 + 17/8
-17/8 = -2 (x - (3/4))^2
17/16 = (x - 3/4)^2
+/- sqrt(17)/4 = x - 3/4

x = [3 +/- sqrt(17)]/4

Therefore, two more points would be ([3 + sqrt(17)]/4, 0) and
([3 - sqrt(17)]/4, 0).

So right now we have a total of 4 points:

(3/4, 17/8)
(0,1)
([3 + sqrt(17)]/4, 0)
([3 - sqrt(17)]/4, 0)

And since we need a 5th point, all we need to do is just plug in any value for x to get y. Let's choose x = 1.
y = -2(1)^2 + 3(1) + 1 = -2 + 3 + 1 = 2

(1,2)

Answer 2

(1)
x² - x - 462

Step 1: First we rewrite the equation in the form ax² + bx = c by adding 462 to each side of the equation. Thus we obtain

x² - x = 462

Step 2: We want to complete the square of x² - x. That is, we want to add a constant term to x² -x so that we get a perfect square trinomial. We do this by taking half the coefficient of x and squaring it.

(-½)² = ¼

Adding ¼ to both sides gives us...

x² - x + ¼ = 462¼ --- Multiply everything by 4 (this step isn't nessecary, but it gets rid of all the fractions)...
4x² - 4x + 1 = 1849
(2x - 1)² = 1849
(2x - 1) = ±√1849
2x - 1 = ± 43 -- Add 1 to both sides...
2x = 1 ± 43 --- Divide both sides by 2...
x = (1 ± 43) / 2

ANSWER: The roots are 22 and (-21).
CHECK:
x = 22
(22)² - (22) = 462
484 - 22 = 462
462 = 462 --- Answer checks out...

x = (-21)
(-21)² - (-21) = 462
441 + 21 = 462
462 = 462 --- Answer checks out...

(2)
y = -2x² + 3x + 1 --- Stick numbers in for x, and solve for y...
x = 0, y = 1
x = 1, y = 2
x = -1, y = -4
x = 2, y = -1
x = -2, y = -13

ANSWER: {(0,1), (1, 2), (-1, -4), (2, -1), (-2, -13)
CHECK: Just plug in x and y...

Answer 3

1) X^2- X- 462= x^2 - x +1/4 -1/4 -462 =(x-1/2)^2 -462.25

2) the coordinates of 5 points, including the vertex and intercept. of y= -2x^2 + 3x+1
(0,1)
(1,-2+3+1) = (1,2)
(-1, -2-3 +1) = ( -1,-4)
(2, -2(4) +3(2)+1) =( 2, -1)
(-2,-2(4) + 3(-2) +1 )= ( -2, -13) .