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lys-0071 [83]
3 years ago
13

Factor the expression below.

Mathematics
1 answer:
Anna35 [415]3 years ago
4 0

Answer:

The answer is (2x-1)(2x+1)

Step-by-step explanation:

4x²-1 can be factorised using difference if two squares

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How to find the vertex calculus 2What is the vertex, focus and directrix of x^2 = 6y
son4ous [18]

Solution:

Given:

x^2=6y

Part A:

The vertex of an up-down facing parabola of the form;

\begin{gathered} y=ax^2+bx+c \\ is \\ x_v=-\frac{b}{2a} \end{gathered}

Rewriting the equation given;

\begin{gathered} 6y=x^2 \\ y=\frac{1}{6}x^2 \\  \\ \text{Hence,} \\ a=\frac{1}{6} \\ b=0 \\ c=0 \\  \\ \text{Hence,} \\ x_v=-\frac{b}{2a} \\ x_v=-\frac{0}{2(\frac{1}{6})} \\ x_v=0 \\  \\ _{} \\ \text{Substituting the value of x into y,} \\ y=\frac{1}{6}x^2 \\ y_v=\frac{1}{6}(0^2) \\ y_v=0 \\  \\ \text{Hence, the vertex is;} \\ (x_v,y_v)=(h,k)=(0,0) \end{gathered}

Therefore, the vertex is (0,0)

Part B:

A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (directrix)

Using the standard equation of a parabola;

\begin{gathered} 4p(y-k)=(x-h)^2 \\  \\ \text{Where;} \\ (h,k)\text{ is the vertex} \\ |p|\text{ is the focal length} \end{gathered}

Rewriting the equation in standard form,

\begin{gathered} x^2=6y \\ 6y=x^2 \\ 4(\frac{3}{2})(y-k)=(x-h)^2 \\ \text{putting (h,k)=(0,0)} \\ 4(\frac{3}{2})(y-0)=(x-0)^2 \\ Comparing\text{to the standard form;} \\ p=\frac{3}{2} \end{gathered}

Since the parabola is symmetric around the y-axis, the focus is a distance p from the center (0,0)

Hence,

\begin{gathered} Focus\text{ is;} \\ (0,0+p) \\ =(0,0+\frac{3}{2}) \\ =(0,\frac{3}{2}) \end{gathered}

Therefore, the focus is;

(0,\frac{3}{2})

Part C:

A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (directrix)

Using the standard equation of a parabola;

\begin{gathered} 4p(y-k)=(x-h)^2 \\  \\ \text{Where;} \\ (h,k)\text{ is the vertex} \\ |p|\text{ is the focal length} \end{gathered}

Rewriting the equation in standard form,

\begin{gathered} x^2=6y \\ 6y=x^2 \\ 4(\frac{3}{2})(y-k)=(x-h)^2 \\ \text{putting (h,k)=(0,0)} \\ 4(\frac{3}{2})(y-0)=(x-0)^2 \\ Comparing\text{to the standard form;} \\ p=\frac{3}{2} \end{gathered}

Since the parabola is symmetric around the y-axis, the directrix is a line parallel to the x-axis at a distance p from the center (0,0).

Hence,

\begin{gathered} Directrix\text{ is;} \\ y=0-p \\ y=0-\frac{3}{2} \\ y=-\frac{3}{2} \end{gathered}

Therefore, the directrix is;

y=-\frac{3}{2}

3 0
1 year ago
Plz help simplify 8/10​
Free_Kalibri [48]

\frac{8}{10} → \frac{4}{5}

8 and 10 can both be divided by 2. So, 8 ÷ 2 is 4 and 10 ÷ 2 is 5.

Hope this helps :)

3 0
3 years ago
Read 2 more answers
Which of the following does NOT have a value of -5?
Contact [7]

Answer:

J. -13-18

Step-by-step explanation:

J will give the value -31

4 0
3 years ago
Simplify 7(-7-6x) using the distributive property
m_a_m_a [10]
You have to distribute the 7 so the final answer will be -49-42x
8 0
3 years ago
If the area of a rectangle is 162 square meters, what is the length of the base
Contact [7]

Answer:

If the long side is down then the length of the base is 81 sq.m.

If the short side is down then the length of the base is 20.25 sq.m.

Step-by-step explanation:

162/4 is 40.5 that means if this were a square it would be 40.5 sq.m. On each side, if the base is longer you multiply 40.5 by 2 (or whatever youre supposed to honestl) and if the base is shorter you divide by 2 (or whatever you were supposed to)

7 0
3 years ago
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