Solving Equations

Equations are an integral part of our lives. We use them not only to compute in various areas of science and mathematics, but also to compute price, tax, debt, interest, etc. \(“\, 1 + 1 = 2"\) is a classic example of an equation, but what exactly is an equation?

An equation is a logical statement stating that two things are equal. An equation contains either terms or expressions. In mathematics, an equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equation true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions. An equation differs from an identity in that an equation is not necessarily true for all possible values of the variable.

The "=" ("equals") symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

There are many types of equations, and they are found in many areas of mathematics. The techniques used to examine them differ according to their type.

Algebra studies two main families of equations: polynomial equations, and among them, linear equations. Polynomial equations have the form \(p(x) = 0\), where \(p\) is a polynomial. Linear equations have the form \(a(x)+b = 0\), where \(a\) is a linear function and \(b\) is a vector. To solve them, one uses algorithmic or geometric techniques, coming from linear algebra or mathematical analysis. Changing the domain of a function can change the problem considerably. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

In geometry, equations are used to describe geometric figures. As equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of being given the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.

Differential equations are equations involving one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model real-life processes in areas such as physics, chemistry, biology, and economics.

Equations can be classified according to the types of operations and quantities involved. Important types include the following:

• An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by their degrees:

  • linear equation for degree 1

  • quadratic equation for degree 2

  • cubic equation for degree 3

  • quartic equation for degree 4

  • quintic equation for degree 5

  • sextic equation for degree 6

  • septic equation for degree 7.

• A Diophantine equation is an equation where the unknowns are required to be integers.

• A transcendental equation is an equation involving a transcendental function of its unknowns.

• A parametric equation is an equation for which the solutions are sought as functions of some other variables, called parameters appearing in the equations.

• A functional equation is an equation in which the unknowns are functions rather than simple quantities.

• A differential equation is a functional equation involving derivatives of the unknown functions.

• An integral equation is a functional equation involving the antiderivatives of the unknown functions.

• An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives of the unknown functions.

• A difference equation is an equation where the unknown is a function \(f,\) which occurs in the equation through \(f(x), f(x−1), \ldots , f(x-k)\) for some positive integers, where \(k\) is called the order of the equation. If \(x\) is restricted to be an integer, a difference equation is the same as a recurrence relation.

First, we will discuss what terms and expressions are used, and give some examples.

The following identities are useful to know:

• \((a+b)^2 = a^2+2ab+b^2\)
• \((a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)\)
• \((a+b)^3 = a^3+b^3+3ab(a+b)\)
• \(a^3+b^3 = (a+b)\big(a^2-ab+b^2\big)\)
• \((a-b)^2 = a^2-2ab+b^2\)
• \((a-b+c)^2 = a^2+b^2+c^2+2(-ab-bc+ca)\)
• \((a+b-c)^2 = a^2+b^2+c^2+2(ab-bc-ca)\)
• \((-a+b+c)^2 = a^2+b^2+c^2+2(-ab+bc-ca)\)
• \((a-b-c)^2 = a^2+b^2+c^2+2(-ab+bc-ca)\)
• \((-a+b-c)^2 = a^2+b^2+c^2+2(-ab-bc+ca)\)
• \((-a-b+c)^2 = a^2+b^2+c^2+2(ab-bc-ca)\)
• \((a-b)^3 = a^3-b^3-3ab(a-b)\)
• \(a^3-b^3 = (a-b)\big(a^2+ab+b^2\big)\)
• \(a^2-b^2 = (a+b)(a-b)\)
• \((a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4\)
• \((a-b)^4 = a^4-4a^3b+6a^2b^2-4ab^3+b^4\)
• \(a^4+b^4 = \big(a^2+b^2+ \sqrt2ab\big)\big(a^2+b^2- \sqrt2ab)\)
• \(a^4-b^4 = (a+b)(a-b)\big(a^2+b^2\big)\)

If \(x + 5 = 19 \), find \(x\).

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We have

\[\begin{align} x + 5 &= 19\\ x &= 19 - 5\\
&= 14.\ _\square \end{align}\]

If \(5x + 9x = 16 - 2x\), find \(x\).

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We have

\[\begin{align} 14x &= 16 -2x\\ 14x + 2x &= 16 \\ 16x &= 16 \\ x &= \frac{16}{16}\\ &= 1.\ _\square \end{align}\]

If \(\frac{x^2}{x-3} = 4x,\) find \(x\).

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We have

\[\begin{align} \frac{x^2}{x-3} &= 4x \\ x^2 &= 4x(x-3) \\ x^2 &= 4x^2 - 12x \\ x^2 - 4x^2 &= -12x \\ -3x^2 &= -12x \\ x^2 &= 4x \\ x^2 - 4x &= 0 \\ x(x-4) &= 0 \\ \Rightarrow x&=0 \ \ \text{or} \ \ x=4.\ _\square \end{align}\]

Solve for \(x:\)

\[ x^2+3x+2=0. \]

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We have

\[\begin{align} x^2+3x+2&=0 \\ x^2+2x+x+2&=0\\ x(x+2)+1(x+2)&=0\\ (x+2)(x+1)&=0\\ x+2&=0 \ \ \text{or} \ \ x+1=0 \\ \Rightarrow x&=-2 \ \ \text{or} \ \ x=-1.\ _\square \end{align}\]

Solve \[3X^4+6X^3-123X^2-126X+1080=0.\]

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Quartic equations are solved in several steps.

First, we simplify the equation by dividing all terms by \(a,\) the leading coefficient, so the equation then becomes \[X^4+2X^3-41X^2-42X+360=0,\] where  \(a=1, b=2, c=-41, d=-42,\) and \(e=360.\)

Next, we define the variable \(f:\) \[f = c - \frac {3b^2}{8}.\] Plugging in the numbers from the above equation, we get \[f = -41 - \frac {3\times 2\times 2}{8}=-42.5.\] Next we define \(g:\) \[g = d + \frac {b^3}{8} - \frac{b\times c}{2}.\] Plugging in the numbers, we get \[g = -42 + 1 - \frac{2 \times (-41)}{2}=0.\] Next, we define \(h:\) \[h = e - \frac{3\times b^4}{256} + \frac{b^2\times c}{16} - \frac {b\times d}{4}.\] Plugging in the numbers, we get \(h = 370.5625.\)

Next, we plug the numbers \(f, g, h\) into the following cubic equation: \[\begin{align} Y^3 + \frac{f}{2}\times Y^2 + \frac{f^2 -4h}{16} \times Y - \frac{g^2}{64} &= 0\\ Y^3 -21.25 Y^2 + \frac{1806.25 -(4 \times 370.5625)}{16} \times Y - \frac{0^2}{64} &= 0\\ Y^3 -21.25 Y^2 + \frac{(1806.25 -1482.25)}{16} \times Y -0&= 0\\ Y^3 -21.25 Y^2 + 20.25 Y &= 0\\ Y(Y-1)(Y-20.25)&=0. \end{align}\] Then the 3 roots of the equation are \[Y_1= 0, \quad Y_2= 1, \quad Y_3= 20.25.\] Now, let \(p\) and \(q\) be the square roots of any 2 non-zeroes of \(Y_1, Y_2,\) or \(Y_3:\) \[p=\sqrt{20.25} = 4.5, \quad q=\sqrt{1}= 1.\] Then \[r= \frac{-g}{8\times pq} = 0, \quad s=\frac{b}{4\times a}= \frac {6}{4×3} = 0.5.\] Then the four roots of the quartic equation are  \[\begin{align} X_1&= p + q + r -s = 4.5 + 1 + 0 - 0.5 = 5\\ X_2&= p - q - r -s = 4.5 - 1 - 0 - 0.5 = 3\\ X_3&= -p + q - r -s = -4.5 + 1 - 0 - 0.5 = -4\\ X_4&= -p - q + r -s = -4.5 - 1 + 0 - 0.5 = -6.\ _\square \end{align} \]