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Math III · Mathematics

Equations · Equations

Solving Other Types of Equations

You can solve $x^2 - 5x + 6 = 0$ in your sleep. But what about $x^4 - 5x^2 + 4 = 0$, or $\sqrt{x+7} = x + 1$, or $2^x = 5$? None of them looks like a quadratic, yet each one is a quadratic (or a linear equation) in disguise. In this section you learn to spot the disguise, to remove it with the right trick, and to pay the "tax" that some tricks charge: checking for solutions that don't really work.

Equations quadratic in form

Why does this work? Because $x^4 = (x^2)^2$. If you call $x^2$ by a new name, $t$, the fourth power becomes a square and the equation looks just like the ones you know.

The same idea works for other exponents. For $x^6 - 7x^3 - 8 = 0$, use $t = x^3$: $t^2 - 7t - 8 = (t - 8)(t + 1) = 0$, so $x^3 = 8$ or $x^3 = -1$, which gives $x = 2$ or $x = -1$. Here a negative $t$ is fine, because a cube can be negative.

Quadratic-in-form equations often come from geometry.

Practice★☆☆

Solve the following biquadratic equation by finding all its real solutions: $x^4 - 13x^2 + 36 = 0$

Solve the following biquadratic equation by finding all its real solutions:

$x^4 - 13x^2 + 36 = 0$

Practice★★☆

The diagonal of a rectangle measures $10$ in and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the length of its sides and solve it.

The diagonal of a rectangle measures $10$ in and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the length of its sides and solve it.

Polynomial equations of degree 3 or higher

There is no simple "cubic formula" you can use in class, so you factor. The plan: put everything on one side so the other side is $0$, find one zero, divide it out, and keep going until you reach factors of degree 1 or 2.

The graph of $y = x^3 - 2x^2 - 5x + 6$ crosses the x-axis at each real solution of $x^3 - 2x^2 - 5x + 6 = 0$. Let's find them exactly.

Polynomial equations also come from word problems and from building a polynomial backwards from its zeros.

A box from a sheet of cardboard

An open-top box is made from a $10$ in. by $8$ in. sheet by cutting a square of side $x$ from each corner and folding up the sides. For what $x$ is the volume $48$ in.³?

The base is $(10 - 2x)$ by $(8 - 2x)$ and the height is $x$, so $x(10 - 2x)(8 - 2x) = 48$. Expand: $4x^3 - 36x^2 + 80x - 48 = 0$, and dividing by $4$: $x^3 - 9x^2 + 20x - 12 = 0$.

Try divisors of $12$. For $x = 1$: $1 - 9 + 20 - 12 = 0$ ✓. Synthetic division by $x - 1$ gives coefficients $1,\ -8,\ 12$, so the quotient is $x^2 - 8x + 12 = (x - 2)(x - 6)$. The zeros are $x = 1, 2, 6$.

Apply the domain. Both $10 - 2x$ and $8 - 2x$ must be positive, so $0 < x < 4$. That rules out $x = 6$ but keeps $x = 1$ and $x = 2$. Check $x = 2$: $2 \cdot 6 \cdot 4 = 48$ ✓. Both $x = 1$ and $x = 2$ work.

Practice★☆☆

Factor the polynomial and solve the cubic equation: $x^3 - 4x^2 - x + 4 = 0$

Factor the polynomial and solve the cubic equation:

$x^3 - 4x^2 - x + 4 = 0$

Practice★★☆

A box without a lid is to be made from a rectangular piece of cardboard of $20 \times 15$ in, by cutting out equal squares of side $x$ from the corners and folding up…

A box without a lid is to be made from a rectangular piece of cardboard of $20 \times 15$ in, by cutting out equal squares of side $x$ from the corners and folding up the edges. 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.

Practice★★★

A card PIN number is made up of four positive, consecutive integer digits in increasing order. If we form a monic polynomial $P(x)$ of degree $4$ whose roots are exactly…

A card PIN number is made up of four positive, consecutive integer digits in increasing order. If we form a monic polynomial $P(x)$ of degree $4$ whose roots are exactly those four digits, its constant term is equal to $24$. Determine the expanded polynomial $P(x)$ and the four-digit PIN number.

Rational equations

A rational equation has the unknown in a denominator. The trick is to clear the fractions by multiplying by the least common denominator (LCD), which leaves a polynomial equation.

The solutions of $\dfrac{5}{x} - \dfrac{6}{x^2} = 1$ are the x-coordinates where the curve meets the line $y = 1$.

Practice★☆☆

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:

$\frac{x}{x-2} + \frac{3}{x+2} = \frac{8}{x^2-4}$

Practice★☆☆

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:

$\frac{2}{x} + \frac{1}{x^2} = 3$

Rational equations in word problems

Two kinds of problems lead to rational equations.

Work problems. If a job takes $x$ hours alone, you do $\dfrac{1}{x}$ of the job per hour. When two workers (or pipes) work together, their rates add.
Travel problems. Time $= \dfrac{\text{distance}}{\text{speed}}$, so a time difference gives an equation with fractions.

A road trip across Texas

Mateo drives $180$ miles at a constant speed. If he had driven $15$ mph faster, the trip would have taken $1$ hour less. Find his speed.

Organize the data. Let $v$ be his speed in mph.

	Distance	Speed	Time
Actual trip	$180$	$v$	$\dfrac{180}{v}$
Faster trip	$180$	$v + 15$	$\dfrac{180}{v+15}$

The actual time is $1$ hour longer: $\dfrac{180}{v} - \dfrac{180}{v+15} = 1$.

Multiply by $v(v + 15)$: $180(v + 15) - 180v = v(v + 15)$, so $2700 = v^2 + 15v$, so $v^2 + 15v - 2700 = 0$, so $(v - 45)(v + 60) = 0$. A speed must be positive, so $v = 45$ mph. Check: $\dfrac{180}{45} = 4$ hours and $\dfrac{180}{60} = 3$ hours ✓.

Practice★★☆

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.

Practice★★★

A cyclist travels from town A to town B, separated by $60$ miles, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ mph, so it…

A cyclist travels from town A to town B, separated by $60$ miles, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ mph, so it takes $1$ hour more to complete the same route. Set up the rational equation that models the situation and determine the outward speed.

Radical equations

In a radical equation (some problems say irrational equation) the unknown is under a root. You get rid of the root by raising both sides to a power, but that move can create fake solutions, so a final check is mandatory.

When there are two radicals, isolate one of them, square, and you will have one radical left.

Radicals also appear when you use the Pythagorean Theorem to write a distance.

A cable laid from the river bank

A tower $T$ stands on a straight river bank. A cabin $C$ is $4$ miles downstream from the tower and $2$ miles inland. A cable runs from a point $P$ on the bank to the tower, and another from $P$ to the cabin. If the total cable is $5$ miles, how far from the tower is $P$? (Assume $P$ is between the tower and the point on the bank directly across from the cabin.)

Let $x$ be the distance $TP$. The second cable is the hypotenuse of a right triangle with legs $4 - x$ and $2$, so its length is $\sqrt{(4 - x)^2 + 4}$. The total length gives $x + \sqrt{(4 - x)^2 + 4} = 5$.

Isolate the radical: $\sqrt{(4 - x)^2 + 4} = 5 - x$. Square both sides: $x^2 - 8x + 20 = x^2 - 10x + 25$. The $x^2$ terms cancel, leaving $2x = 5$, so $x = 2.5$.

Check: $5 - x = 2.5 \ge 0$ ✓, and $2.5 + \sqrt{1.5^2 + 4} = 2.5 + \sqrt{6.25} = 2.5 + 2.5 = 5$ ✓. Point $P$ is $2.5$ miles from the tower.

Practice★☆☆

Find the real solutions of the irrational equation, checking their validity: $\sqrt{2x+5} - x = 1$

Find the real solutions of the irrational equation, checking their validity:

$\sqrt{2x+5} - x = 1$

Practice★★☆

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:

$\sqrt{x+4} + \sqrt{x-1} = 5$

Practice★★★

A telecommunications tower is located on the straight bank of a river. A cable is to be laid in a straight line from a point $P$ on the bank to the tower, and another…

A telecommunications tower is located on the straight bank of a river. A cable is to be laid in a straight line from a point $P$ on the bank to the tower, and another cable from $P$ to a cabin located $5$ miles downstream from the tower and $1$ miles inland. If the total length of the cable used is $6$ miles, set up an irrational equation and determine at what distance from the tower point $P$ is located.

Absolute value equations

The absolute value $|A|$ is the distance from $A$ to $0$ on the number line. (Some problems call these modulus equations.) Distance is never negative, which controls everything.

If the expression inside is itself a quadratic, split first and then solve each quadratic.

Nested absolute values are peeled from the outside in.

Practice★☆☆

Find the values of $x$ that satisfy the modulus equation: $|2x - 5| = 7$

Find the values of $x$ that satisfy the modulus equation:

$|2x - 5| = 7$

Practice★★☆

Find all the real values of $x$ that satisfy the following quadratic modulus equation: $|x^2 - 5x + 4| = 2$

Find all the real values of $x$ that satisfy the following quadratic modulus equation:

$|x^2 - 5x + 4| = 2$

Practice★★★

Find all the real solutions of the nested modulus equation: $||x - 2| - 3| = 1$

Find all the real solutions of the nested modulus equation:

$||x - 2| - 3| = 1$

Exponential equations

In an exponential equation the unknown is in an exponent. Here are the main strategies.

When the bases differ, factor the common power on each side.

Practice★☆☆

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:

$3^{2x+1} - 27 = 0$

Practice★★☆

A bacteria culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation…

A bacteria culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation $4^x - 5 \cdot 2^x + 4 = 0$. Find the values of $x$ by applying a change of variable.

Practice★★★

Solve the following exponential equation involving different bases, by taking out a common factor 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 a common factor on each side:

$2^{x+1} + 2^{x-1} = 3^{x} + 3^{x-2}$

When the bases don't match: logarithms

Sometimes you can't rewrite both sides with the same base. For example, how long does it take $\$1{,}000$ to grow to $\$1{,}500$ at $5\%$ interest? You need the inverse of an exponential: the logarithm.

Practice★★☆

A sample of radioactive material decays such that the remaining amount $M(t)$ in ounces after $t$ years obeys the expression $M(t) = 200 \cdot e^{-0.05t}$. How many…

A sample of radioactive material decays such that the remaining amount $M(t)$ in ounces after $t$ years obeys the expression $M(t) = 200 \cdot e^{-0.05t}$. How many years must pass for exactly $50$ ounces of material to remain? Express the exact result using natural logarithms.

Logarithmic equations

In a logarithmic equation the unknown is inside a logarithm. Your goal: get a single logarithm on each side, then either convert to exponential form or equate the arguments.

Logarithms also let you compare sizes on a scale. The energy $E$ of an earthquake satisfies $\log E = 11.8 + 1.5M$, where $M$ is the Richter magnitude. Suppose quake 2 releases $10^k$ times as much energy as quake 1. Then $\log E_2 = k + \log E_1$, because $\log(10^k E_1) = k + \log E_1$. Subtracting the two magnitude equations gives $1.5(M_2 - M_1) = k$.

Practice★☆☆

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:

$\log_2(x+3) + \log_2(x-3) = 4$

Practice★★☆

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:

$\log(x+1) - \log(x-1) = \log(x+4) - \log(x+1)$

Practice★★★

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?

Equations with a parameter

Sometimes an equation contains a letter such as $k$, and the question is how many real solutions it has for each value of $k$. Reduce to a quadratic with a substitution, then analyze the roots with the discriminant and with the sum and product of the roots.

Slide $k$ and count how many times the graph of $y = x^4 - 4x^2 + k$ touches or crosses the x-axis. That is the number of real solutions. Let's explain what you see.

How many real solutions does $x^4 - 4x^2 + k = 0$ have?

Let $t = x^2$: $t^2 - 4t + k = 0$. The discriminant is $\Delta = 16 - 4k$. The roots satisfy $t_1 + t_2 = 4$ and $t_1 t_2 = k$.

Each positive $t$ gives two values of $x$ (namely $\pm\sqrt{t}$), $t = 0$ gives one ($x = 0$), and a negative $t$ gives none. So count the positive roots.

Go case by case:

$k$	Roots in $t$	Real solutions in $x$
$k > 4$	none ($\Delta < 0$)	$0$
$k = 4$	$t = 2$ (double)	$2$ ($x = \pm\sqrt2$)
$0 < k < 4$	two positive roots ($t_1 t_2 > 0$, $t_1 + t_2 > 0$)	$4$
$k = 0$	$t = 0$ and $t = 4$	$3$ ($x = 0, \pm 2$)
$k < 0$	one positive, one negative ($t₁ t₂ < 0$)	$2$

Practice★★★

Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Justify your answer.

Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Justify your answer.

Where two graphs meet

A system with two curves, such as $y = \sqrt{2x + 3}$ and $y = x$, asks where the graphs intersect. Because both equations say "$y = $", set the right sides equal. That gives one equation in $x$ of a type you already know. Then substitute back to find $y$ and check the point in both original equations.

Practice★★★

Solve the following non-linear system that models the intersection of two trajectories in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of…

Solve the following non-linear system that models the intersection of two trajectories in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of intersection algebraically.

Back to the start: $x^4 - 5x^2 + 4 = 0$ is a quadratic in $t = x^2$, $\sqrt{x + 7} = x + 1$ becomes a quadratic once you square, and $2^x = 5$ is solved by a logarithm. The disguise is always removed by a substitution, a power, or a log, and the price is the same every time: undo the substitution, then check the answers against the original equation.

Summary

Summary

Quadratic in form ($ax^4 + bx^2 + c = 0$): let $t = x^2$, solve for $t$, then use $x = \pm\sqrt{t}$. Discard $t < 0$.
Polynomials of degree 3+: list possible zeros with the Rational Root Theorem, divide with synthetic division, repeat on the quotient. Write $0$ for missing terms. For a monic $P$ with zeros $r_i$: $P(x) = \prod (x - r_i)$.
Rational equations: factor denominators, multiply by the LCD, solve, reject restricted values. In word problems, rates add ($\frac{1}{x}$ of the job per hour).
Radical equations: isolate the radical, raise to the power, repeat if needed, and check every candidate in the original equation.
Absolute value: $|A| = k$ means $A = k$ or $A = -k$ (none if $k < 0$). Nested bars: peel the outer layer first.
Exponential: same base, equate exponents. Repeated power, use $u = a^x$ (and $u > 0$). Different bases, factor or use $\ln$: $b^x = c \Rightarrow x = \dfrac{\ln c}{\ln b}$.
Logarithmic: combine into one log per side, convert with $\log_b N = p \iff N = b^p$, and check that all arguments are positive.
With a parameter or in a system: reduce first, count the valid roots (signs of sum and product), and check each candidate in the original problem.

Let's practice! 8 medium exercises in this section

★★☆ The diagonal of a rectangle measures $10$ in and its area is $48\text{ cm}^2$. Set up a…★★☆ Two hoses fill a tank in $6$ hours working together. If the first hose takes $5$ hours…★★☆ Solve the following irrational equation with two radicals and check the solution…

Let's practice! 8 hard exercises in this section

★★★ Determine for which values of the parameter $k$ the biquadratic equation…★★★ A cyclist travels from town A to town B, separated by $60$ miles, at a constant speed. On…★★★ A telecommunications tower is located on the straight bank of a river. A cable is to be…

See all the exercises in this section