Hi. I need help with these questions.
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Answer 20 and 21

20.  The vertex is (2/3, 14/3) | p = 3, q = -2/3 and r = 14/321.   20x² + 2x - 3 = 0

Step-by-step explanation:

20.

Giveny = 3x² - 4x + 6To findThe least value of the y and the corresponding value of xConstants p, q and r such that 3x² - 4x + 6 = p(x + q)² + rSolution

The given is the parabola with positive a coefficient, so it opens up and the minimum point its vertex.

The vertex has x = -b/2a and corresponding y- coordinate is found below:

x = - (- 4)/2*3 = 2/3, and y = 3(2/3)² - 4(2/3) + 6 = 4/3 - 8/3 + 6 = 14/3So the vertex is (2/3, 14/3)

The vertex form of the line has the equation:

y = a(x - h)² + k, where (h, k) is the vertex

Plugging in the values:

y = 3(x - 2/3)² + 14/3

Comparing with p(x + q)² + r, to find out that:

p = 3, q = -2/3 and r = 14/3

21.

(i) α and β are the roots of: ax² + bx + c = 0

Show that:

α + β = -b/a and αβ = c/aSolution

Knowing the roots, put the equation as:

(x - α)(x - β) = 0x² - αx - βx + αβ = 0x² - (α+β)x + αβ = 0

Comparing this with the standard form:

ax² + bx + c = 0

Divide by a to make the constants of x² same:

x² + b/ax + c/a = 0

Now comparing the constants:

- (α+β) = b/a ⇒ α+β = - b/aαβ = c/a

(ii)

Givenα and β are the roots of: 3x² - x - 5 = 0To Find The equation with roots 1/2α and 1/2βSolution

The sum and the product of the roots:

α + β = -b/a = 1/3αβ = c/a = -5/3

The equation is:

(x - 1/2α)(x - 1/2β) = 0x² - (1/2α + 1/2β)x + 1/(2α)(2β) = 0x² - (α + β)/(2αβ)x + 1/4αβ = 0x² - (1/3)/(2(-5/3))x + 1/(4(-5/3)) = 0x² + 1/10x - 3/20 = 020x² + 2x - 3 = 0

The correct answer is -4