First- and second-degree equations
Sharing out an inheritance, working out the ages of two siblings or finding the moment when a model rocket reaches a certain height: all of these problems become a first- or second-degree equation. Mastering both, with and without denominators, with all their terms or with some missing, is the basis of almost all Bachillerato algebra.
First-degree equations with denominators
When there are denominators, multiply the whole equation by the lowest common multiple of all of them, which removes the fractions in a single step.
A content creator on a popular social network observes that the number of new followers gained each day follows a linear trend. If the number of followers $x$ satisfies…
A content creator on a popular social network observes that the number of new followers gained each day follows a linear trend. If the number of followers $x$ satisfies the following equation, how many followers has he gained today?
$3(x - 15) + 2x = 2(x + 30)$
In designing a green area, a landscape architect wants to create a rectangular flowerbed whose base is 4 metres longer than its height. If the total area of the…
In designing a green area, a landscape architect wants to create a rectangular flowerbed whose base is 4 metres longer than its height. If the total area of the flowerbed must be $96 \text{ m}^2$, set up a quadratic equation and calculate the exact dimensions of the base and height.
Quadratic equation: general form and formula
A photography studio wants to print square posters for an exhibition. If twice the area of one of these posters is equal to $50$ $dm^2$, calculate the length of the side…
A photography studio wants to print square posters for an exhibition. If twice the area of one of these posters is equal to $50$ $dm^2$, calculate the length of the side $x$ of the poster by solving the equation:
$2x^2 = 50$
Solve the following linear equation, paying special attention to calculating the lowest common multiple and the effect of negative signs in front of fractions:…
Solve the following linear equation, paying special attention to calculating the lowest common multiple and the effect of negative signs in front of fractions:
$$\frac{2x - 3}{4} - \frac{x - 5}{6} = \frac{5x - 1}{3} - 2$$Incomplete equations: solving without the general formula
Find the real solutions to the following complete quadratic equation using the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $x^2 - x - 6 = 0$
Find the real solutions to the following complete quadratic equation using the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:
$x^2 - x - 6 = 0$
Find a quadratic equation of the form $ax^2 + bx + c = 0$ whose solutions are $x_1 = 3$ and $x_2 = -4$.
Find a quadratic equation of the form $ax^2 + bx + c = 0$ whose solutions are $x_1 = 3$ and $x_2 = -4$.
The discriminant and Vieta's formulas
A sports centre offers a monthly membership. If 20 more people enrol than expected, the cost per person would decrease by $4 \text{ €}$ to maintain the same total…
A sports centre offers a monthly membership. If 20 more people enrol than expected, the cost per person would decrease by $4 \text{ €}$ to maintain the same total revenue of $2400 \text{ €}$. Set up the corresponding equation to find out how many people were initially expected.
A digital marketing agency has shown that the success of a social media campaign (measured in thousands of "likes") follows a rational model depending on time $x$ (in…
A digital marketing agency has shown that the success of a social media campaign (measured in thousands of "likes") follows a rational model depending on time $x$ (in weeks) according to the following expression:
$$\frac{2x+5}{x-1} - \frac{x-2}{x+3} = \frac{20}{x^2+2x-3}$$Find the value of $x$ that satisfies this equation. Do not forget to check whether the values obtained are valid within the domain of the given rational expressions.
Real-life problems
A courier company charges a fixed fee of $5 \text{ €}$ for shipping a package, plus an additional cost for each kilometre of distance. If for a delivery to a city…
A courier company charges a fixed fee of $5 \text{ €}$ for shipping a package, plus an additional cost for each kilometre of distance. If for a delivery to a city located $120 \text{ km}$ away they charged $23 \text{ €}$:
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Determine the linear equation relating the total cost ($C$) to the distance ($d$).
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Calculate how much it will cost to send a package over a distance of $350 \text{ km}$.
Solve the following equation with nested radicals and make sure to verify the validity of the solutions obtained in the original expression: $$\sqrt{2x-1} + \sqrt{x+4} =…
Solve the following equation with nested radicals and make sure to verify the validity of the solutions obtained in the original expression:
$$\sqrt{2x-1} + \sqrt{x+4} = 6$$