Quadratic Equations
Mateo launches a model rocket in a field. Its height in feet after $t$ seconds is $h = 48t - 16t^2$. At what times is the rocket exactly 32 ft high? A rocket goes up and comes back down, so it passes through the same height twice. Quadratic equations are the tool for questions like this one, and we'll answer it in the word problems section.
What is a quadratic equation?
The graph of $y = ax^2+bx+c$ is a parabola. Solving the equation means finding where the parabola crosses the $x$-axis. Move the sliders and watch how the number of $x$-intercepts changes.
Rewrite this one in standard form, then come back and finish it once you've learned the methods below.
Expand the algebraic identities, group like terms, and solve the following equation: $(x + 2)^2 - 2(x - 1)^2 = 3x + 6$
Expand the algebraic identities, group like terms, and solve the following equation:
$(x + 2)^2 - 2(x - 1)^2 = 3x + 6$
Special cases: no $x$ term or no constant
Some problems call these incomplete quadratic equations. They have a missing term, and you don't need the quadratic formula for them.
Solve the following incomplete quadratic equation by isolating the unknown directly: $4x^2 - 64 = 0$
Solve the following incomplete quadratic equation by isolating the unknown directly:
$4x^2 - 64 = 0$
Calculate the solutions of the following incomplete quadratic equation by factoring out the common factor: $3x^2 + 15x = 0$
Calculate the solutions of the following incomplete quadratic equation by factoring out the common factor:
$3x^2 + 15x = 0$
Factoring and the zero product property
When a quadratic factors, the work is nearly done. Set one side equal to $0$, factor, and use the zero product property.
Slide $r$ and $s$ and notice that the roots of $(x - r)(x - s)$ are exactly $r$ and $s$.
You can also run the process backwards to build an equation from its solutions.
Find the solutions of the following equation which is already factored, without needing to expand the product: $(2x - 5)(x + 4) = 0$
Find the solutions of the following equation which is already factored, without needing to expand the product:
$(2x - 5)(x + 4) = 0$
Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing that its only solutions are $x_1 = 3$ and $x_2 = -8$.
Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing that its only solutions are $x_1 = 3$ and $x_2 = -8$.
The quadratic formula
Not every quadratic factors nicely. The quadratic formula works for every quadratic equation.
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:
$x^2 - 7x + 10 = 0$
The discriminant: how many solutions?
The part under the square root, $b^2 - 4ac$, tells you what kind of solutions you'll get before you finish solving.
Determine whether the equation $2x^2 + 3x + 5 = 0$ has a solution 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 a solution 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 real double solution.
Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single real double solution.
Using one solution to find the other
If you know one solution, you can learn a lot without solving the whole equation. There are two tools: substituting the known solution, and the sum and product of the roots.
Knowing that one of the solutions of the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without fully solving the equation the value of the other solution $x_2$…
Knowing that one of the solutions of the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without fully solving the equation the value of the other solution $x_2$ and the value of the coefficient $b$.
Equations with a parameter
Sometimes an equation has an extra letter, a parameter such as $m$ or $k$. You are asked how many real solutions there are for each value of the parameter. The discriminant is the key, with one extra warning: if $a$ depends on the parameter, check separately the value that makes $a = 0$. At that value the equation is no longer quadratic.
Discuss, in terms of the different values of the parameter $m$, the number of real solutions that the following quadratic equation has: $mx^2 - 2(m-1)x + m = 0$ Make…
Discuss, in terms of the different values of the parameter $m$, the number of real solutions that the following quadratic equation has:
$mx^2 - 2(m-1)x + m = 0$
Make sure to analyze what happens when the discriminant is positive, zero, or negative, and do not forget the special case in which the leading coefficient is zero.
Word problems with quadratics
Quadratic models appear when something is multiplied by itself or by a related quantity: areas, projectile motion, profit. Use the same plan every time.
Back to Mateo's rocket. Here is the graph of its height together with the line $h = 32$.
Back to Mateo's rocket: it really does reach 32 ft twice, once on the way up and once on the way down. The peak is at $t = 1.5$ s, halfway between the two answers.
The square of a natural number minus three times the number is exactly equal to $18$. Set up a quadratic equation and calculate what the number is.
The square of a natural number minus three times the number is exactly equal to $18$. Set up a quadratic equation and calculate what the number is.
An aerospace engineer launches a test rocket. The height $h$ (in feet) reached by the rocket in terms of the time $t$ (in seconds) is given by the expression…
An aerospace engineer launches a test rocket. The height $h$ (in feet) reached by the rocket in terms of the time $t$ (in seconds) is given by the expression $h(t) = 40t - 5t^2$. Determine at what times the rocket is exactly at a height of $75\text{ ft}$.
A company that manufactures electric scooters has determined that its daily profits $B$ (in hundreds of dollars) depend on the number of scooters manufactured $x$…
A company that manufactures electric scooters has determined that its daily profits $B$ (in hundreds of dollars) depend on the number of scooters manufactured $x$ according to the equation $B(x) = -x^2 + 50x - 400$. Calculate exactly how many scooters they must manufacture for the profit to be zero (the company's break-even point).
Geometry problems
Draw the figure, label the sides with the same variable, and use an area or Pythagorean formula.
To renovate the rectangular living room of a house, the architect indicates that the length measures $2\text{ m}$ more than the width. If the total area of the living…
To renovate the rectangular living room of a house, the architect indicates that the length measures $2\text{ m}$ 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.
In a skating ramp shaped like a right triangle, the longer leg measures $7\text{ m}$ more than the shorter leg, and the hypotenuse measures $2\text{ m}$ more than the…
In a skating ramp shaped like a right triangle, the longer leg measures $7\text{ m}$ more than the shorter leg, and the hypotenuse measures $2\text{ m}$ more than the longer leg. Set up a quadratic equation using the Pythagorean theorem and calculate the length of the three sides of the ramp.
A rectangular photograph with dimensions $20\text{ cm}$ base by $15\text{ cm}$ height is going to be framed with a mat of uniform width $x$. If the total area of the set…
A rectangular photograph with dimensions $20\text{ cm}$ base by $15\text{ cm}$ height is going to be framed with a mat of uniform width $x$. If the total area of the set (photograph plus mat) must be exactly $500\text{ cm}^2$, set up a quadratic equation and calculate the width $x$ of the mat.
Problems with a formula or a ratio
Sometimes the problem hands you a formula and a value, and you must solve for the input. Other times it gives a ratio. Write the ratio as two equal fractions and cross-multiply: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.
The number of diagonals of a polygon with $n$ sides is $\frac{n(n-3)}{2}$. The graph below shows this together with the line at $20$ diagonals.
In a volleyball league, each team plays exactly one game against all the others. Knowing that the total number of games played in a tournament of $n$ teams is given by…
In a volleyball league, each team plays exactly one game against all the others. Knowing that the total number of games played in a tournament of $n$ teams is given by the formula $\frac{n(n-1)}{2}$, and that at the end of the season a total of $105$ games have been played, set up the corresponding equation and find out how many teams participated in the league.
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 this 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 $x$.
Equations with fractions
Some problems call these algebraic fractions. In the US they are rational equations. To solve one, clear the denominators and you often land on a quadratic.
Solve the following equation by first working with algebraic fractions. Make sure to check the solutions obtained to rule out possible extraneous solutions that make the…
Solve the following equation by first working with algebraic fractions. Make sure to check the solutions obtained to rule out possible extraneous solutions that make the denominators zero:
$\frac{x}{x-2} + \frac{2x}{x+2} = \frac{16}{x^2-4}$
Two hoses of different thicknesses 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…
Two hoses of different thicknesses 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 thick hose to fill the tank on its own. Set up a quadratic equation by adding the fractions of the tank that each hose fills in one hour, and calculate how long each hose would take working individually.
A group of students charters a bus for a trip for a total cost of $\$400\text{ \$}$. At the last minute, $5$ more students sign up for the trip, which means that each of…
A group of students charters a bus for a trip for a total cost of $\$400\text{ \$}$. At the last minute, $5$ more students sign up for the trip, which means that each of those who were initially going has to pay $\$4\text{ \$}$ less. Set up a quadratic equation to calculate how many students were going at the beginning and solve it.
Radical equations
When the variable is under a square root, isolate the radical, then square both sides. That usually gives a quadratic. Squaring can create extra answers, so you must check.
The two graphs above, $y = \sqrt{x + 6}$ and $y = x$, meet at only one point, even though squaring gives two candidates.
Solve the following equation in which a radical appears, by first isolating the root and squaring both sides. It is mandatory to check the solutions obtained in the…
Solve the following equation in which a radical appears, by first isolating the root and squaring both sides. It is mandatory to check the solutions obtained in the original equation to rule out possible extraneous solutions:
$\sqrt{2x + 7} - x = 2$
Equations that are quadratic in form
Some equations have $x^4$ and $x^2$ and no other powers. Some problems call them biquadratic equations. In the US we say they are quadratic in form. Substitute $u = x^2$ and you get an ordinary quadratic in $u$.
The area of a rectangle is $60\text{ cm}^2$ and the length of its diagonal is $13\text{ cm}$. Calling the base $x$ and the height $y$, 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 the base $x$ and the height $y$, express $y$ in terms of $x$ using the area data. Next, substitute into the Pythagorean theorem for the diagonal and solve the resulting biquadratic equation to find the exact dimensions of the rectangle.