Alan

"The line y= 3x-p+1 is a tangent to the parabola y= x²

Evaluate p "

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The ladder, wall and ground form a right-angled triangle so you can use Pythagoras to find the height, h:

h^2 + 1.5^2 = 6.8^2

h = sqrt(6.8^2-1.5^2) m ≈ 6.6 m

General equation for line in slope intercept form is:  y = mx + c  where m is slope and c is y-intercept.

If line is parallel to y = 4x + 7 then it has the same slope, so m = 4

If it passes through (0, 8), the y-intercept is 8  (i.e. it's the value of y when x is 0), so c = 8

Hence y = 4x + 8

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With "at least" questions it is often worth thinking about "none" first!

The probability of drawing no Aces from 3 random picks: p_none =  (48/52)*(47/51)*(46/50) →4324/5525

The probability of at least one Ace:   p_{at least one} = 1 - p_none → 1 - 4324/5525 → 1201/5525 ≈ 0.217

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Like so:

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Follow the procedure shown at: https://web2.0calc.com/questions/if-we-express-3x-2-6x-2-in-the-form-a-x-h-2-k-what-is-a-h-k

See https://web2.0calc.com/questions/if-we-express-3x-2-6x-2-in-the-form-a-x-h-2-k-what-is-a-h-k#r4

Well, we have:

coeff of x^2:   a = 3

coeff of x:      -2ah = -6  → -2*3*h = -6  → 6h = 6  →  h = 1

constant;      ah^2 + k = -2 →  3*(1^2) + k = -2  →  3 + k = -2  →  k = -5

Hence  a + h + k = 3 + 1 - 5  or  a + h + k = -1

Same process as for https://web2.0calc.com/questions/the-quadratic-x-2-20x-36-can-be-written-in-the-form, except you now compare the coefficient of the x^2 term as well.

a(x - h)^2 + k → a(x^2 - 2hx + h^2) + k → ax^2 - 2ahx + ah^2 +k

coeff of x^2:   a = 3

coeff of x:      -2ah = -6

constant;      ah^2 + k = -2

I'll leave you to complete it!

"The quadratic x^2-20x+36 can be written in the form (x+b)^2+c where b and c are constants.  What is b+c"

Expand (x+b)^2 + c → x^2 + 2bx + b^2 + c

Compare with x^2 - 20x + 36

Coefficient of x term:    2b = -20  hence b = -10

Constant term:    b^2 + c = 36,    (-10)^2 + c = 36,    100 + c = 36,   c = -64

b + c = -74