Quadratic Equations
A test rocket goes up and comes back down, and someone asks: when is it exactly 75 m high? There is not just one answer: there is one moment on the way up and another on the way down. Quadratic equations describe exactly this kind of situation. On this page you review how to solve them and learn how to get information from them without fully solving them. At the end we return to the rocket.
General form and quadratic formula
Move the sliders and notice that the solutions are the points where the parabola crosses the $x$-axis.
Apply the quadratic formula to find the roots of the following complete quadratic equation: $x^2 - 7x + 10 = 0$
Apply the quadratic formula to find the roots of the following complete quadratic equation:
$x^2 - 7x + 10 = 0$
Incomplete equations
Solve the following incomplete quadratic equation by rearranging directly for the unknown: $4x^2 - 64 = 0$
Solve the following incomplete quadratic equation by rearranging directly for the unknown:
$4x^2 - 64 = 0$
Find the solutions to the following incomplete quadratic equation by taking out a common factor: $3x^2 + 15x = 0$
Find the solutions to the following incomplete quadratic equation by taking out a common factor:
$3x^2 + 15x = 0$
Factoring and perfect square trinomials
The quadratic formula always works, but sometimes there are shorter paths. If the left side factors, we again use the fact that a product is zero when some factor is zero.
Another method is completing the square. Remember that $(x+h)^2 = x^2 + 2hx + h^2$: if you have $x^2+bx$, you need to add $\left(\frac{b}{2}\right)^2$ to make it a perfect square. In fact, this is where the quadratic formula comes from.
Find the solutions to the following equation, which is already factorised, without expanding the product: $(2x - 5)(x + 4) = 0$
Find the solutions to the following equation, which is already factorised, without expanding the product:
$(2x - 5)(x + 4) = 0$
The discriminant
Determine whether the equation $2x^2 + 3x + 5 = 0$ has solutions in the set of real numbers by first calculating the value of its discriminant $\Delta$.
Determine whether the equation $2x^2 + 3x + 5 = 0$ has solutions in the set of real numbers by first calculating the value of its discriminant $\Delta$.
Find the value of the coefficient $c$ in the equation $x^2 + 4x + c = 0$ so that it has a single repeated real solution.
Find the value of the coefficient $c$ in the equation $x^2 + 4x + c = 0$ so that it has a single repeated real solution.
Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single repeated real solution.
Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single repeated real solution.
Sum and product of the roots
It also works in reverse: to build an equation from its solutions.
Write down a quadratic equation in the form $x^2 + bx + c = 0$ given that its only solutions are $x_1 = 3$ and $x_2 = -8$.
Write down a quadratic equation in the form $x^2 + bx + c = 0$ given that its only solutions are $x_1 = 3$ and $x_2 = -8$.
Given that one of the solutions to the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without solving the equation completely the value of the other solution…
Given that one of the solutions to the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without solving the equation completely the value of the other solution $x_2$ and the value of the coefficient $b$.
Equations with a parameter
Sometimes the equation has an extra letter, a parameter ($m$ or $k$), and you are asked to study how many solutions there are depending on its value. The discriminant is still the tool, but if the coefficient of $x^2$ depends on the parameter you must separately check when it equals $0$: then the equation stops being quadratic.
Discuss, in terms of the different values of the parameter $m$, the number of real solutions to the following quadratic equation: $mx^2 - 2(m-1)x + m = 0$ Make sure to…
Discuss, in terms of the different values of the parameter $m$, the number of real solutions to the following quadratic equation:
$mx^2 - 2(m-1)x + m = 0$
Make sure to analyse what happens when the discriminant is positive, zero or negative, and do not forget the special case where the leading coefficient is zero.
Equations that reduce to quadratics
Many equations do not start out quadratic, but become one after operating on them. With special products, it is enough to expand and group terms. With fractions or radicals, extraneous solutions can appear when operating, so you always check.
A biquadratic equation such as $x^4-ax^2+b=0$ is solved with the substitution $u=x^2$: it becomes $u^2-au+b=0$. At the end, you go back to $x=\pm\sqrt{u}$ for each positive $u$.
Expand the algebraic identities, collect like terms and solve the following equation: $(x + 2)^2 - 2(x - 1)^2 = 3x + 6$
Expand the algebraic identities, collect like terms and solve the following equation:
$(x + 2)^2 - 2(x - 1)^2 = 3x + 6$
Solve the following equation by first operating with algebraic fractions. Make sure to check the solutions obtained to rule out any false roots that make the…
Solve the following equation by first operating with algebraic fractions. Make sure to check the solutions obtained to rule out any false roots that make the denominators zero:
$\frac{x}{x-2} + \frac{2x}{x+2} = \frac{16}{x^2-4}$
Solve the following equation containing a radical, first isolating the square root and squaring both sides. You must check the solutions obtained in the original…
Solve the following equation containing a radical, first isolating the square root and squaring both sides. You must check the solutions obtained in the original equation to rule out any false roots:
$\sqrt{2x + 7} - x = 2$
Problems that lead to a quadratic
We return to the rocket: its height is $h(t)=40t-5t^2$.
To set up word problems: choose the unknown, write the relationship (area, Pythagorean theorem, sum of work fractions, cost split among people), bring it to $ax^2+bx+c=0$, solve, and discard the solutions that make no sense (negative lengths or numbers of people, for example).
The square of a natural number minus three times the number is exactly equal to $18$. Set up a quadratic equation and find the number.
The square of a natural number minus three times the number is exactly equal to $18$. Set up a quadratic equation and find the number.
To renovate the rectangular living room of a house, the architect states that the length is $2\text{ m}$ more than the width. If the total area of the living room is…
To renovate the rectangular living room of a house, the architect states that the length is $2\text{ m}$ more than the width. If the total area of the living room is $35\text{ m}^2$, set up a quadratic equation and calculate the exact dimensions of the room.
An aerospace engineer launches a test rocket. The height $h$ (in meters) reached by the rocket as a function of time $t$ (in seconds) is given by the expression…
An aerospace engineer launches a test rocket. The height $h$ (in meters) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 40t - 5t^2$. Determine the times at which the rocket is at a height of exactly $75\text{ m}$.
Ticket sales for the school musical yield a profit $B$ (in tens of pesos) that depends on the number of tickets sold $x$ (in tens) according to the equation…
Ticket sales for the school musical yield a profit $B$ (in tens of pesos) that depends on the number of tickets sold $x$ (in tens) according to the equation $B(x) = -x^2 + 50x - 400$. Calculate exactly how many tickets must be sold for the profit to be zero (the break-even point).
On a skateboard ramp in the shape of a right-angled triangle, the longer leg is $7\text{ m}$ longer than the shorter leg, and the hypotenuse is $2\text{ m}$ longer than…
On a skateboard ramp in the shape of a right-angled triangle, the longer leg is $7\text{ m}$ longer than the shorter leg, and the hypotenuse is $2\text{ m}$ longer than the longer leg. Set up a quadratic equation using Pythagoras' theorem and calculate the lengths of the three sides of the ramp.
A rectangular photograph with base dimensions $20\text{ cm}$ by height $15\text{ cm}$ is to be framed with a mount of uniform width $x$. If the total area of the set…
A rectangular photograph with base dimensions $20\text{ cm}$ by height $15\text{ cm}$ is to be framed with a mount of uniform width $x$. If the total area of the set (photograph plus mount) must be exactly $500\text{ cm}^2$, set up a quadratic equation and calculate the width $x$ of the mount.
In a volleyball league, each team plays exactly one match against every other team. Given that the total number of matches played in a tournament of $n$ teams is given…
In a volleyball league, each team plays exactly one match against every other team. Given that the total number of matches played in a tournament of $n$ teams is given by the formula $\frac{n(n-1)}{2}$, and that by the end of the season a total of $105$ matches have been played, set up the corresponding equation and find out how many teams took part in the league.
Two hoses of different thickness fill a water tank together in $6\text{ hours}$. If only one of them were used, the thinner hose would take $5\text{ hours}$ longer than…
Two hoses of different thickness fill a water tank together in $6\text{ hours}$. If only one of them were used, the thinner hose would take $5\text{ hours}$ longer than the thicker hose to fill the tank on its own. Set up a quadratic equation by adding the fractions of the tank filled by each hose in one hour, and calculate how long each hose would take working individually.
A sculptor wants to divide a brass rod of length $1\text{ metro}$ into two pieces of lengths $x$ and $1-x$ (where $x$ is the larger piece, that is, $x > 1-x$). They wish…
A sculptor wants to divide a brass rod of length $1\text{ metro}$ into two pieces of lengths $x$ and $1-x$ (where $x$ is the larger piece, that is, $x > 1-x$). They wish to do it in such a way that the ratio of the whole rod to the larger piece is equal to the ratio of the larger piece to the smaller piece. Set up the corresponding equation and find the exact value of $x$.
The area of a rectangle is $60\text{ cm}^2$ and the length of its diagonal is $13\text{ cm}$. Calling $x$ the base and $y$ the height, express $y$ in terms of $x$ using…
The area of a rectangle is $60\text{ cm}^2$ and the length of its diagonal is $13\text{ cm}$. Calling $x$ the base and $y$ the height, express $y$ in terms of $x$ using the given area. Then, substitute into Pythagoras' theorem for the diagonal and solve the resulting biquadratic equation to find the exact dimensions of the rectangle.
A group of students hires a bus for a trip at a total cost of $\$8\,000$. At the last minute, $5…
A group of students hires a bus for a trip at a total cost of $\$8\,000$. At the last minute, $5$ more students join the trip, which means that each of those who were originally going pays $\$80$ less. Set up a quadratic equation to calculate how many students were originally going, and solve it.