**Answer 1**

buddy no one is going to help you with all of that.... one question is already plenty but your too greedy.... sorry bub.

**Answer 2**x^2 - 4x + 4 = 0

(x)^2 - 2(x)(2) + (2)^2 = 0

(x - 2)^2 = 0

THus x = +2, and x = +2.

**Answer 3**x² - 4x + 4

If this is a perfect square polynomial, the factors are two identical binomials: x² - 4x + 4 = (x - 2)(x - 2) = (x - 2)²

Completing the square

x² + 16x

To generate missing term: start with linear term (16x), take half of coefficient (8) and square it (64)

x² + 16x + 64

Now that you have a perfect square polynomial, it factors to identical binomials: x² + 16x + 64 = (x + 8)(x + 8) = (x + 8)²

x² - x + 2 Calculate the discriminant

With expression in standard form (ax² + bx + c), discriminant is the stuff under the radical in the quadratic formula: b² - 4ac.

If discriminant is positive, there are two real roots. If zero, one real root. If negative, two complex roots.

**Answer 4**Well i have nothing to do at work. Lazy afternoon, so i'll oblige.....

1. x2 - 4x + 4 = 0

=> x2 - 2*2x + 4 = 0

=> (x-2)^2 =0 (perfect sq.)

=> x-2 = 0

=> x = 2

2. x2 - 5x + 6 = 0

=> x2 - 2*2.5x + 6 = 0

first two terms indicate it can be perfect square of x-2.5 provided we have the third term 6.25. So add and subtract 0.25 (so there's no net change)

=> (x2 - 2*2.5x + 6) + 0.25 - 0.25 = 0

=> (x2 - 2*2.5x + 6.25) - 0.25 = 0

=> (x - 2.5)^2 = 0.25

=> (x - 2.5) = +/- 0.5

=> x = 2.5+0.5 or 2.5-0.5

=> x = 3 or 2 (two unique solutions)

3. x2 + 16x

= x2 + 2*8x

if we add 64, it will be a perfect square: (x+8)^2

4. x^2 - 8x + 4 = 0

=> x2 - 2*4x + 4 = 0

To make perfect square, third term must be 16. (see 2 above). Add and subtract 12

=> x2 - 2*4x + 4 + 12 - 12 =0

=> x2 - 2*4x + 16 = 12

=> (x-4)^2 = 12

=> x-4 = +/- sqrt(12)

=>x = 4 +/- 2sqrt(3)

=> x = 4 + 2sqrt(3) or x = 4 - 2sqrt(3)

5. 6x2 + 12x = 0

Divide both sides by 6

=> x2 + 2x = 0

=> x(x+2) = 0

=> (x-0)(x+2) = 0

=> x = 0 or x = -2

6. 2x2 + 2x - 1 = 0

comparing with standard quadratic equation Ax2 + Bx + C,

A = 2, B = 2, C = -1

Discriminant, D = (B2 - 4AC)

= (12)

solution using quadratic formula,

x = [-B + sqrt(D)] / (2A) or x = [-B - sqrt(D)] / (2A)

=> x = [-2+2sqrt(3)] / 4 or x = [-2-2sqrt(3)] / 4

=> x = (1/2) + (sqrt(3)/2) or x = (1/2) - (sqrt(3)/2)

7. x2 - x + 2

comparing with standard quadratic equation Ax2 + Bx + C,

A = 1, B = -1, C = 2

Discriminant, D = (B2 - 4AC)

= (1 - 4*1*2)

= - 7

8.3x2 - 6x + 1 = 0

comparing with standard quadratic equation Ax2 + Bx + C,

A = 3, B = -6, C = 1

Discriminant, D = (B2 - 4AC)

= (36 - 4*3*1)

= 24

Since discriminant is non-negative, it will have real roots

Since discriminant is non-zero, it will have two distinct real roots

9. y = -2x2 + x +3

discriminant = 1 - 4*-2*3 = 1+24 = 25

Since discriminant is non-zero, the equation has 2 real roots. At the roots, the right hand side is zero, or Y = 0. Thus the roots are the points at which the parabola intersects the x-axis.

Since there are 2 roots, the parabola has 2 points in common with the x-axis.

Also, since the the coefficient of x^2 is negative the parabola opens downwards. We know that it intersects the x-axis at 2 points. Thus the vertex MUST lie above the x-axis

10. Using similar approach as 9,

discriminant = 144 - 4*3*12 = 0

Since discriminant is zero, the equation has ONE real root, or one point in common with the x-axis. To maintain parabolic symmetry, this point HAS TO BE the vertex.

Thus, the vertex lies ON the x-axis

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YOU OWE ME BIGTIME!!!!