Solving Equations | Brilliant Math & Science Wiki (2024)

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 byRobert 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.

Algebrastudies two main families of equations: polynomial equations,and among them,linear equations. Polynomial equations have the form \(p(x)=0\), where\(p\)is apolynomial. Linear equations have the form\(a(x)+b=0\), where\(a\)is alinear functionand\(b\)is avector. To solve them, one uses algorithmic or geometric techniques, coming from linear algebraormathematical analysis. Changing thedomain of a functioncan change the problem considerably. Algebra also studiesDiophantine equationswhere the coefficients and solutions are integers. The techniques used are different and come fromnumber 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.

Ingeometry, equations are used to describe geometric figures. As equations that are considered, such asimplicit equationsorparametric 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 ofalgebraic geometry, an important area of mathematics.

Differential equationsare equations involving one or more functions and their derivatives. They aresolved 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 operationsand quantities involved. Important types include the following:

• Analgebraic equationorpolynomialequation is an equation in which both sides are polynomials (see alsosystem of polynomial equations). These are further classified bytheir degrees:

  • linear equation for degree 1

  • quadratic equationfor degree 2

  • cubic equationfor degree 3

  • quartic equationfor degree 4

  • quintic equationfor degree 5

  • sextic equation for degree 6

  • septic equationfor degree 7.

• ADiophantine equationis an equation where the unknowns are required to beintegers.

• Atranscendental equationis an equation involving atranscendental functionof its unknowns.

• Aparametric equationis an equation for which the solutions are sought as functions of some other variables, calledparametersappearing in the equations.

• Afunctional equationis an equation in which the unknowns arefunctionsrather than simple quantities.

• Adifferential equationis a functional equation involvingderivativesof the unknown functions.

• Anintegral equationis a functional equation involving theantiderivativesof the unknown functions.

• Anintegro-differential equationis a functional equation involving both thederivativesand the antiderivativesof the unknown functions.

• Adifference equationis 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 theorderof 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 \(x\).

We have

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

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

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 \(x\).

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 \(x:\)

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

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

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\[Y_1= 0, \quadY_2= 1, \quad Y_3=20.25.\]Now, let \(p\) and \(q\) be the square roots of any 2non-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\[\begin{align}X_1&= p + q + r -s = 4.5 + 1 + 0 - 0.5 = 5\\X_2&= p - q - r -s = 4.5 - 1 - 0 - 0.5 = 3\\X_3&= -p + q - r -s = -4.5 + 1 - 0 - 0.5 = -4\\X_4&= -p - q + r -s = -4.5 - 1 + 0 - 0.5 = -6.\ _\square\end{align} \]