Question

Hi. I need help with these questions.
See image for question.
Answer 20 and 21

Answer 1

• 20. The vertex is (2/3, 14/3) | p = 3, q = -2/3 and r = 14/3
• 21. 20x² + 2x - 3 = 0

Explanation:

20.

Given

• y = 3x² - 4x + 6

To find

• The least value of the y and the corresponding value of x
• Constants p, q and r such that 3x² - 4x + 6 = p(x + q)² + r

Solution

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 = 14/3

• So 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

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21.

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

Show that:

• α + β = -b/a and αβ = c/a

Solution

Knowing the roots, put the equation as:

• (x - α)(x - β) = 0
• x² - αx - βx + αβ = 0
• x² - (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/a
• αβ = c/a

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(ii)

Given

• α and β are the roots of: 3x² - x - 5 = 0

To Find

• The equation with roots 1/2α and 1/2β

Solution

The sum and the product of the roots:

• α + β = -b/a = 1/3
• αβ = c/a = -5/3

The equation is:

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