Other Equations
Yaritza deposits $\text{RD}\$50{,}000$ in a credit union that pays her 8% annual compound interest. How many years have to go by for her money to double? The unknown is in the exponent, and neither factoring nor the quadratic formula works right away. In this section you will see that, with the right trick (matching bases, a change of variable, a logarithm, or synthetic division), many "strange" equations reduce to something you already know. We answer Yaritza's question in the applications section.
Exponential equations
The solution process depends on how the equation looks:
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Same base on both sides: write both sides as powers of that base and set the exponents equal (if $a^m=a^n$, then $m=n$).
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A repeated power with related exponents ($9^x$ and $3^x$, for example): change of variable $u=a^x$ and solve the equation in $u$.
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Sums of powers with the same base: factor out the power with the smallest exponent.
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Bases that cannot be matched: take the logarithm of both sides and bring down the exponent: $\log a^x=x\log a$.
Why does it work? The function $a^x$ (with $a>0$, $a\neq1$) never repeats values, so two equal powers of the same base force the exponents to be equal.
The graph shows why $3^x=20$ has only one solution: the curve $y=3^x$ crosses the line $y=20$ once, between $x=2$ and $x=3$.
Determine the value of $x$ that satisfies the following exponential equation: $3^{2x+1} - 27 = 0$
Determine the value of $x$ that satisfies the following exponential equation:
$3^{2x+1} - 27 = 0$
A bacterial culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation…
A bacterial culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation $4^x - 5 \times 2^x + 4 = 0$. Find the values of $x$ by applying a substitution.
Solve the following exponential equation involving different bases by taking out common factors on each side: $2^{x+1} + 2^{x-1} = 3^{x} + 3^{x-2}$
Solve the following exponential equation involving different bases by taking out common factors on each side:
$2^{x+1} + 2^{x-1} = 3^{x} + 3^{x-2}$
Logarithmic equations
The process:
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Determine the domain: every argument must be $>0$.
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Using the properties, leave a single logarithm on each side:
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If you get $\log_a N=\log_a M$, set the arguments equal: $N=M$. If you get $\log_a N=p$, use the definition: $N=a^p$.
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Solve the resulting equation.
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Check each solution against the domain.
Why? The logarithm does not repeat values, so equating logarithms of the same base lets you equate the arguments. And $\log_aN=p$ means exactly $a^p=N$.
Apply the properties of logarithms to solve the equation: $\log_2(x+3) + \log_2(x-3) = 4$
Apply the properties of logarithms to solve the equation:
$\log_2(x+3) + \log_2(x-3) = 4$
Apply the properties of logarithms to group the terms and solve the equation: $\log(x+1) - \log(x-1) = \log(x+4) - \log(x+1)$
Apply the properties of logarithms to group the terms and solve the equation:
$\log(x+1) - \log(x-1) = \log(x+4) - \log(x+1)$
Applications: growth, interest, and scales
Exponentials and logarithms model bacterial growth, compound interest, the decay of a material, or the energy of an earthquake. To solve for an unknown in the exponent you use the logarithm, and with $e$ you use the natural logarithm: $\ln e^x=x$. To get the final numerical value, use a calculator or a technology tool, and first leave the exact result (for example, $t=\dfrac{\ln 5}{0.2}$).
Back to the beginning: Yaritza's money doubles in about 9 years.
A sample of radioactive material decays such that the remaining amount $M(t)$ in grams after $t$ years obeys the expression $M(t) = 200 \times e^{-0.05t}$. How many…
A sample of radioactive material decays such that the remaining amount $M(t)$ in grams after $t$ years obeys the expression $M(t) = 200 \times e^{-0.05t}$. How many years must pass for exactly $50$ grams of material to remain? Express the exact result using natural logarithms.
The energy $E$ (in ergs) released by an earthquake is related to its magnitude $M$ on the Richter scale by the equation $\log E = 11.8 + 1.5M$. If an earthquake releases…
The energy $E$ (in ergs) released by an earthquake is related to its magnitude $M$ on the Richter scale by the equation $\log E = 11.8 + 1.5M$. If an earthquake releases $1000$ times more energy than another whose magnitude is $4.2$, what is the magnitude of the first earthquake on the Richter scale?
Polynomial equations of degree greater than 2
To solve an equation like $x^3-7x+6=0$ you factor the polynomial using synthetic division (Ruffini's rule) and apply the fact that a product is zero when some factor is zero. It is the direct application of what you already know about the factor theorem.
The graph of $x^3-4x^2-x+4$ crosses the $x$-axis at $-1$, $1$, and $4$: its three roots.
Factorise the polynomial and solve the cubic equation: $x^3 - 4x^2 - x + 4 = 0$
Factorise the polynomial and solve the cubic equation:
$x^3 - 4x^2 - x + 4 = 0$
An open-topped box is to be constructed from a rectangular piece of cardboard measuring $20 \times 15$ cm, by cutting out equal squares of side length $x$ from the…
An open-topped box is to be constructed from a rectangular piece of cardboard measuring $20 \times 15$ cm, by cutting out equal squares of side length $x$ from the corners and folding up the sides. If the volume of the box must be $378\text{ cm}^3$, set up the corresponding cubic polynomial equation and find the value of $x$, knowing that it is an integer.
A card's PIN consists of four positive, consecutive integers in increasing order. If we construct a monic polynomial $P(x)$ of degree $4$ whose roots are precisely those…
A card's PIN consists of four positive, consecutive integers in increasing order. If we construct a monic polynomial $P(x)$ of degree $4$ whose roots are precisely those four digits, its constant term turns out to be equal to $24$. Determine the expanded polynomial $P(x)$ and the four-digit PIN.
Rational equations
The process: (1) find the values that make some denominator zero (they cannot be solutions); (2) multiply the whole equation by the common denominator; (3) solve the polynomial equation; (4) discard the forbidden solutions.
Find the value of $x$ that satisfies the following equation with the unknown in the denominator: $\frac{x}{x-2} + \frac{3}{x+2} = \frac{8}{x^2-4}$
Find the value of $x$ that satisfies the following equation with the unknown in the denominator:
$\frac{x}{x-2} + \frac{3}{x+2} = \frac{8}{x^2-4}$
Solve the following rational equation by reducing it to a polynomial equation: $\frac{2}{x} + \frac{1}{x^2} = 3$
Solve the following rational equation by reducing it to a polynomial equation:
$\frac{2}{x} + \frac{1}{x^2} = 3$
Two hoses fill a tank in $6$ hours working together. If the first hose takes $5$ hours less than the second to fill the tank on its own, how long does each one take? Set…
Two hoses fill a tank in $6$ hours working together. If the first hose takes $5$ hours less than the second to fill the tank on its own, how long does each one take? Set up and solve the corresponding rational equation.
A cyclist travels from town A to town B, separated by $60$ km, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ km/h, taking…
A cyclist travels from town A to town B, separated by $60$ km, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ km/h, taking $1$ hour longer to complete the same journey. Set up the rational equation that models the situation and determine the speed on the outward journey.
Biquadratic equations
Solve the following biquadratic equation, finding all its real solutions: $x^4 - 13x^2 + 36 = 0$
Solve the following biquadratic equation, finding all its real solutions:
$x^4 - 13x^2 + 36 = 0$
The diagonal of a rectangle measures $10$ cm and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the lengths of its sides and solve it.
The diagonal of a rectangle measures $10$ cm and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the lengths of its sides and solve it.
Irrational equations
The process: (1) leave one radical alone on one side; (2) square both sides; (3) if another radical remains, repeat; (4) solve; (5) always check in the original equation.
Why must you check? When you square, $a=b$ becomes $a^2=b^2$, which also holds if $a=-b$. Then false solutions appear.
Find the real solutions to the irrational equation, checking their validity: $\sqrt{2x+5} - x = 1$
Find the real solutions to the irrational equation, checking their validity:
$\sqrt{2x+5} - x = 1$
Solve the following irrational equation with two radicals and check the solution obtained: $\sqrt{x+4} + \sqrt{x-1} = 5$
Solve the following irrational equation with two radicals and check the solution obtained:
$\sqrt{x+4} + \sqrt{x-1} = 5$
A telecommunications tower is located on the straight bank of a river. A cable is to be stretched in a straight line from a point $P$ on the bank to the tower, and…
A telecommunications tower is located on the straight bank of a river. A cable is to be stretched in a straight line from a point $P$ on the bank to the tower, and another cable from $P$ to a cabin located $5$ km downstream from the tower and $1$ km inland. If the total length of cable used is $6$ km, set up a radical equation and determine the distance from point $P$ to the tower.
Absolute value equations
The absolute value $|a|$ is the distance from $a$ to 0, and $|x-2|=4$ means "$x$ is 4 units from 2."
Find the values of $x$ that satisfy the equation with absolute value: $|2x - 5| = 7$
Find the values of $x$ that satisfy the equation with absolute value:
$|2x - 5| = 7$
Find all real values of $x$ satisfying the following equation with quadratic modulus: $|x^2 - 5x + 4| = 2$
Find all real values of $x$ satisfying the following equation with quadratic modulus:
$|x^2 - 5x + 4| = 2$
Find all real solutions of the equation with nested modulus: $||x - 2| - 3| = 1$
Find all real solutions of the equation with nested modulus:
$||x - 2| - 3| = 1$
Mixed equations and equations with a parameter
Some equations mix techniques. For example, a root equal to an absolute value: you square, solve, and check (as in irrational equations).
With a parameter you have to discuss: see for which values there are solutions and how many.
Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Give full justification for your…
Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Give full justification for your answer.
Solve the following non-linear system modelling the intersection of two paths in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of…
Solve the following non-linear system modelling the intersection of two paths in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of intersection algebraically.