Solving Algebraic Problems with Quadratic Equations

by | Jun 20, 2024 | mathematics

 

a) Show that the solutions of the equation \(x^2 – 6x – 43 = 0\) can be written in the form \(x = p \pm \frac{q\sqrt{13}}{13}\), where \(p\) and \(q\) are positive integers.
b) Based on the above or any other method, solve the inequality \(x^2 – 6x – 43 \le 0\).
A function \(f\) is defined by \(f(x) = 3(x-1)^2 – 18\), \(x \in \mathbb{R}\).

a) Write \(f(x)\) in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
b) Find the coordinates of the vertex of the graph of \(f\).
c) Find the equation of the axis of symmetry of the graph of \(f\).
d) Indicate the range of \(f\).
e) The graph of \(f\) is translated by the vector \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\) to form a new curve that represents a new function \(g(x)\).

Find \(g(x)\) in the form \(px^2 + qx + r\), where \(p\), \(q\), and \(r\) are constants.
a) Solve the equation \(8x^2 + 6x – 5 = 0\) by factorization.
b) Determine the range of values of \(k\) for which the equation \(8x^2 + 6x – 5 = k\) has no real solutions.
Consider the function \(f(x) = -x^2 – 10x + 27\), \(x \in \mathbb{R}\).
a) Show that the function \(f\) can be expressed in the form \(f(x) = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants.
b) Based on the above, write the coordinates of the vertex of the graph of \(y = f(x)\).
c) Based on the above, write the equation of the axis of symmetry of the graph of \(y = f(x)\).
The quadratic curve \(y = x^2 + bx + c\) cuts the x-axis at \( (10, 0) \) and has the equation of the line of symmetry \(x = \frac{5}{2}\).

a) Find the values of \(b\) and \(c\).
b) Based on the above, or any other method, find the other two coordinates where the curve intersects the y-axis.
Consider the function \(f(x) = 2x^2 – 4x – 8\), \(x \in \mathbb{R}\).

a) Show that the function \(f\) can be expressed in the form \(f(x) = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants.
b) The function \(f(x)\) can be obtained through a sequence of transformations of \(g(x) = x^2\). Describe each transformation in order.
Consider the equation \(f(x) = 2kx^2 + 6x + k\), \(x \in \mathbb{R}\).

a) For the case where the equation \(f(x) = 0\) has two equal real roots, find the possible values of \(k\).
b) For the case where the equation of the axis of symmetry of the curve \(y = f(x)\) is \(x + 1 = 0\), find the value of \(k\).
c) Solve the equation \(f(x) = 0\) when \(k = 2\).
A curve \(y = f(x)\) passes through the points with coordinates \(A(-12, 10)\), \(B(0, -16)\), \(C(2, 9)\), and \(D(14, -10)\).

a) Write the coordinates of each point after the curve has been transformed by \(f(x) \rightarrow f(2x)\).
b) Write the coordinates of each point after the curve has been transformed by \(f(x) \rightarrow f(-x) + 3\).

 

 

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

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Solving Algebraic Problems with Quadratic Equations

Problem 1:

a)
To show that the solutions of the equation (x^2 – 6x – 43 = 0) can be written in the form (x = p \pm \frac{q\sqrt{13}}{13}), we first find the roots using the quadratic formula:
[ x = \frac{-(-6) \pm \sqrt{(-6)^2 – 4(1)(-43)}}{2(1)} ]
[ x = \frac{6 \pm \sqrt{36 + 172}}{2} ]
[ x = \frac{6 \pm \sqrt{208}}{2} ]
[ x = \frac{6 \pm 4\sqrt{13}}{2} ]
[ x = 3 \pm 2\sqrt{13} ]
Hence, we can write ( x = p \pm \frac{q\sqrt{13}}{13} ) with ( p = 3 ) and ( q = 2 ).

b)
To solve the inequality (x^2 – 6x – 43 \le 0), we find the critical points by setting the expression equal to zero and solving for (x):
[ x^2 – 6x – 43 = 0 ]
[ (x – 9)(x + 3) = 0 ]
This gives us critical points at (x = 9) and (x = -3). We then test the intervals created by these points to determine where the inequality holds true. The solution is ( -3 \le x \le 9 ).

Problem 2:

a)
Given the function ( f(x) = 3(x-1)^2 – 18 ), we expand and simplify to express it in the form ( ax^2 + bx + c ):
[ f(x) = 3(x^2 – 2x + 1) – 18 ]
[ f(x) = 3x^2 – 6x + 3 – 18 ]
[ f(x) = 3x^2 – 6x – 15 ]
Therefore, ( a = 3, b = -6, c = -15 ).

b)
The vertex of a quadratic function in the form ( ax^2 + bx + c ) is given by ( (-\frac{b}{2a}, f(-\frac{b}{2a})) ). Substituting the values, we get the vertex as ( (1, -15) ).

c)
The equation of the axis of symmetry is given by ( x = -\frac{b}{2a} ). Substituting values, we have ( x = \frac{6}{6} = 1 ).

d)
The range of a quadratic function in the form ( ax^2 + bx + c ) where ( a > 0 ) is ( [f(h), +\infty) ), where ( f(h) ) is the minimum value of the function. In this case, since ( a = 3 > 0 ), the range is ( [-15, +\infty) ).

e)
To find the new function ( g(x) ) after translating by the vector ( (2, -1) ), we replace ( x ) with ( x-2 ) and ( y ) with ( y+1 ):
[ g(x) = f(x-2)+1 = 3((x-2)-1)^2 – 18 + 1 = 3(x-3)^2 – 17 ]
Therefore, ( g(x) = 3x^2 – 18x +19 ).

These steps provide solutions to the presented algebraic problems involving quadratic equations and functions.

 

 

 

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