Answer:
- 20. The vertex is (2/3, 14/3) | p = 3, q = -2/3 and r = 14/3
- 21. 20x² + 2x - 3 = 0
Step-by-step explanation:
20.
<h3>Given</h3>
<h3>To find</h3>
- 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
<h3>Solution</h3>
The given is the parabola with positive a coefficient, so it opens up and the minimum point its vertex.
<u>The vertex has x = -b/2a and corresponding y- coordinate is found below: </u>
- 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)
<u>The vertex form of the line has the equation:</u>
- y = a(x - h)² + k, where (h, k) is the vertex
<u>Plugging in the values:</u>
<u>Comparing with p(x + q)² + r, to find out that:</u>
- p = 3, q = -2/3 and r = 14/3
=====================================
21.
(i) α and β are the roots of: ax² + bx + c = 0
<u>Show that:</u>
- α + β = -b/a and αβ = c/a
<h3>Solution</h3>
<u>Knowing the roots, put the equation as:</u>
- (x - α)(x - β) = 0
- x² - αx - βx + αβ = 0
- x² - (α+β)x + αβ = 0
<u>Comparing this with the standard form:</u>
<u>Divide by </u><u>a</u><u> to make the constants of x² same:</u>
<u>Now comparing the constants:</u>
- - (α+β) = b/a ⇒ α+β = - b/a
- αβ = c/a
--------------------------------------------
(ii)
<h3>Given</h3>
- α and β are the roots of: 3x² - x - 5 = 0
<h3>To Find </h3>
- The equation with roots 1/2α and 1/2β
<h3>Solution</h3>
<u>The sum and the product of the roots:</u>
- α + β = -b/a = 1/3
- αβ = c/a = -5/3
<u>The equation is:</u>
- (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
0.035. You add the two numbers together and divide by two
Answer:
We have been given the matrix m as m = [874][205].
We can rewrite this matrix as
\begin{gathered}m=\begin{pmatrix}8 &7&4\\2&0&5 \\\end{pmatrix}\\\end{gathered}m=(827045)
Now, we need to the find m_{12}m12
It means we need to find the entity that is present in the first row and the second column.
From the above matrix, we can see that 7 is in the first row and the second column.
Therefore, we have
m_{12}= 7m12=7
5 Hope I helped. Have a good day
Answer:
Solution of given quadratic equation is
Step-by-step explanation:
The given quadratic equation is
The general form of the quadratic equation is given by
Comparing the general form with the given quadratic equation
The solutions of the quadratic equation is given by
Substitute the values of a, b and c
and
Where i represents iota which means that the given quadratic equation has complex roots.
So the solution of given quadratic equation is
The factored form of the given quadratic equation is