Quadratic Equations
Four friends from a high school in Santiago want to rent a sailboat in Samaná for the weekend. If two of them back out, the rest have to pay more. How many were there at the start? That situation is solved with a quadratic equation, and by the end of this page you will be able to set it up and solve it. Since Secondary Year 2 you have known how to apply the quadratic formula; in Year 6 you use it as a tool: you analyze the roots without solving, relate them to complex numbers, and reduce more complicated equations to it.
General form and formula
Move the sliders and watch where the parabola crosses the $x$-axis: those crossings are the solutions.
Solve the following complete quadratic equation using the quadratic formula: $x^2 - 7x + 10 = 0$
Solve the following complete quadratic equation using the quadratic formula:
$x^2 - 7x + 10 = 0$
Incomplete quadratics: solving without the quadratic formula
Find the solutions to the following incomplete quadratic equation by factorising out a common factor: $3x^2 - 12x = 0$
Find the solutions to the following incomplete quadratic equation by factorising out a common factor:
$3x^2 - 12x = 0$
Solve the following incomplete quadratic equation by solving directly for the unknown: $2x^2 - 18 = 0$
Solve the following incomplete quadratic equation by solving directly for the unknown:
$2x^2 - 18 = 0$
The discriminant and complex roots
When $\Delta<0$, write $\sqrt{\Delta}=\sqrt{-\Delta}\,i$ and the roots are $x=\dfrac{-b}{2a}\pm\dfrac{\sqrt{-\Delta}}{2a}\,i$. They are complex conjugates: they have the same real part and opposite imaginary parts.
Without solving the equation, determine the number of real solutions it has by calculating the value of its discriminant: $5x^2 - 3x + 2 = 0$
Without solving the equation, determine the number of real solutions it has by calculating the value of its discriminant:
$5x^2 - 3x + 2 = 0$
Determine the value (or values) of the parameter $k$ such that the equation $x^2 - kx + 9 = 0$ has a single repeated real solution.
Determine the value (or values) of the parameter $k$ such that the equation $x^2 - kx + 9 = 0$ has a single repeated real solution.
Cardano-Vieta: sum and product without solving
Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and product of its roots using Vieta's formulas, without needing to solve the equation.
Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and product of its roots using Vieta's formulas, without needing to solve the equation.
Equations with a parameter and change of variable
Two situations come up a lot: an equation with an extra letter (a parameter, like $m$ or $k$) for which you are asked to discuss the solutions, and an equation that looks complicated but that a change of variable reduces to a quadratic.
In the parametric equation $2x^2 + mx + 15 = 0$, one of the solutions is known to be $x = 3$. Without using the quadratic formula, calculate the value of the parameter…
In the parametric equation $2x^2 + mx + 15 = 0$, one of the solutions is known to be $x = 3$. Without using the quadratic formula, calculate the value of the parameter $m$ and find the other solution.
Discuss the number of real solutions of the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.
Discuss the number of real solutions of the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.
Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation: $(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation:
$(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Quadratic word problems
To set up a problem: name the unknown, write the relationship as an equation, bring it to the form $ax^2+bx+c=0$, solve it, and discard solutions that make no sense (negative lengths or times, non-integers if integers are required).
Go back to the beginning: the sailboat problem (the exercise below) is solved the same way as the boat example, by setting up the cost per person before and after.
In a physics experiment, a water rocket is launched vertically upward. Its height $h$ in meters above the ground, as a function of time $t$ in seconds, is given by the…
In a physics experiment, a water rocket is launched vertically upward. Its height $h$ in meters above the ground, as a function of time $t$ in seconds, is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine the time or times at which the rocket is at a height of exactly $17$ meters.
A photograph measuring $20 \text{ cm}$ high by $30 \text{ cm}$ wide is to be framed with a wooden frame of uniform width $x$. If the total area of the frame (excluding…
A photograph measuring $20 \text{ cm}$ high by $30 \text{ cm}$ wide is to be framed with a wooden frame of uniform width $x$. If the total area of the frame (excluding the photograph) is $216 \text{ cm}^2$, set up an equation and find the width of the frame.
The product of two consecutive odd integers is $323$. Set up a quadratic equation and solve it to find these two numbers. Is there more than one possible pair?
The product of two consecutive odd integers is $323$. Set up a quadratic equation and solve it to find these two numbers. Is there more than one possible pair?
In a right-angled triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed such that two of its sides lie along the legs and one of its vertices…
In a right-angled triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed such that two of its sides lie along the legs and one of its vertices lies on the hypotenuse. If the area of the rectangle must be exactly $12 \text{ cm}^2$, set up a quadratic equation and find the possible dimensions of the rectangle.
A group of friends decides to rent a sailboat for the weekend for a total cost of $21\,600$ pesos, which they agree to split equally. At the last minute, two of the…
A group of friends decides to rent a sailboat for the weekend for a total cost of $21\,600$ pesos, which they agree to split equally. At the last minute, two of the friends cannot attend, forcing the rest of the group to pay $540$ pesos more each to cover the total cost. Set up an equation that reduces to a quadratic to determine how many friends were in the original group.