12.3k views
4 votes
Hi. I need help with these questions.
See image for question.
Answer 20 and 21

Hi. I need help with these questions. See image for question. Answer 20 and 21-example-1
User Auto
by
8.0k points

1 Answer

2 votes

Answer:

  • 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

=====================================

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

--------------------------------------------

(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

User Pate
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories