Quadratic Equations
A soccer ball is kicked straight up and you want to know when it passes through a certain height. A quadratic equation gives you the answer, and it even tells you, without solving it, how many answers there are. In this section you use it as a tool: you solve it in different ways, connect it to its graph, and apply it to problems.
General form and the quadratic formula
Each solution of the equation is a point where the parabola $y=ax^2+bx+c$ crosses the $x$-axis. Move the sliders and watch how the crossings change.
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:
$x^2 - 7x + 10 = 0$
Completing the square
The quadratic formula doesn't come out of nowhere: it is obtained with a method that works for any quadratic, completing the square. The idea is to turn $x^2+bx$ into a perfect square trinomial $(x+\tfrac{b}{2})^2$ by adding $\left(\tfrac{b}{2}\right)^2$ to both sides.
Incomplete equations
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 the graph
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.
Relationships between roots and coefficients
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.
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.
Equations with a parameter and change of variable
Two situations come up often: an equation with an extra letter (a parameter, like $m$ or $k$) for which you have to decide how many solutions it has; and an equation that looks complicated but, with a change of variable, reduces to a quadratic.
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:
$(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Problems with quadratic equations
The height of a ball or a water rocket is a parabola. Ask about a specific height and you get a quadratic equation.
Inscribed figures and similar triangles
When a rectangle is inscribed in a right triangle, with one corner at the right angle and the opposite vertex on the hypotenuse, a small triangle is left above the rectangle with the same angles as the large one: they are similar. That is why their sides are proportional, and that proportion gives you the second relationship you need to set up the equation.
Notice that here $\Delta=0$, and that is why the rectangle is unique. The largest-area rectangle possible in that triangle has area exactly $12\text{ cm}^2$.
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 $7\,200$ 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 $7\,200$ 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 $180$ 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.
Go back to the beginning: to find out when the ball passes through a certain height, you set $h(t)$ equal to that height and solve the quadratic; the discriminant tells you whether or not the ball reaches that height.