Ximena and Diego are going to paint a wall in Puebla. Working together they finish in 4 hours, and Ximena, working alone, would take 6 hours less than Diego. How long does each one take? That equation has the unknown in the denominator and doesn't look like $ax^2+bx+c=0$. Here you will learn to recognize which trick turns it into something familiar: a change of variable, a property of exponents or logarithms, squaring both sides, or synthetic division.
Biquadratic equations
Why does it work? Because $x^4=(x^2)^2$. If you call $x^2$ by the name $t$, the equation becomes quadratic.
The graph crosses the $x$-axis at four points, which are the four solutions.
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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:
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.
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 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.
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.
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.
When a letter like $k$ appears, you have to discuss how many solutions the equation has depending on its value. With a change of variable, each restriction of the substitution ($t\ge0$) counts.
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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.
To find where two curves cross, you set their expressions equal and solve. Then you have to check each point in both curves.
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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.
Speed problems lead to rational equations with $t=\dfrac dv$: if the time changes by one hour, you set up an equation with two fractions.
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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.
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.
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?