Quadratic Equations
A storage box for video games has a square base with an area of 144 square inches. How long is each side? You need a number $x$ with $x^2 = 144$. That is a quadratic equation, because the unknown is squared. A quadratic equation can have two solutions, one solution, or none. There is also one formula that solves all of them.
Standard Form
Why $a \neq 0$? If $a = 0$, the $x^2$ term disappears and you are left with a linear equation.
The graph of $y = ax^2 + bx + c$ is a parabola. The solutions of $ax^2 + bx + c = 0$ are the points where the parabola crosses the x-axis. Move the sliders and watch how many crossings there are:
For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and indicate whether it is a complete or incomplete equation: a)…
For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and indicate whether it is a complete or incomplete equation:
- a) $x^2 - 5x + 6 = 0$
- b) $2x^2 - 32 = 0$
- c) $x^2 + 8x = 0$
- d) $3 - x^2 + 2x = 0$
Incomplete Equations: $ax^2 + c = 0$
When the middle term is missing, you can solve without any formula. Get $x^2$ alone, then take the square root.
Solve the following incomplete quadratic equations of the type $ax^2 + c = 0$. Remember that if the result of the square root is not a real number, you must indicate it:…
Solve the following incomplete quadratic equations of the type $ax^2 + c = 0$. Remember that if the result of the square root is not a real number, you must indicate it:
- a) $x^2 - 25 = 0$
- b) $3x^2 - 12 = 0$
- c) $x^2 + 16 = 0$
- d) $2x^2 = 50$
The area of the square base of a storage box for video games is $144\text{ cm}^2$. Calculate how much the side of the base measures by setting up a quadratic equation.
The area of the square base of a storage box for video games is $144\text{ cm}^2$. Calculate how much the side of the base measures by setting up a quadratic equation.
Incomplete Equations: $ax^2 + bx = 0$
When the constant term is missing, factor out the GCF (greatest common factor), which here includes $x$. Then use the Zero Product Property: if a product equals $0$, at least one factor is $0$.
If an equation has the $x^2$ term on one side and the $x$ term on the other (like $3x^2 = 6x$), first move everything to one side so that the other side is $0$.
Solve the following incomplete quadratic equations of the type $ax^2 + bx = 0$ by extracting a common factor: a) $x^2 - 7x = 0$ b) $2x^2 + 10x = 0$ c) $3x^2 = 6x$ d)…
Solve the following incomplete quadratic equations of the type $ax^2 + bx = 0$ by extracting a common factor:
- a) $x^2 - 7x = 0$
- b) $2x^2 + 10x = 0$
- c) $3x^2 = 6x$
- d) $x^2 + x = 0$
If we add its triple to the square of an integer, we obtain zero as a result. What numbers could it be? Set up the equation and find the possible solutions.
If we add its triple to the square of an integer, we obtain zero as a result. What numbers could it be? Set up the equation and find the possible solutions.
Solve the following incomplete quadratic equations using the fastest method (extracting a common factor or directly isolating the square root): a) $4x^2 - 100 = 0$ b)…
Solve the following incomplete quadratic equations using the fastest method (extracting a common factor or directly isolating the square root):
- a) $4x^2 - 100 = 0$
- b) $3x^2 + 15x = 0$
- c) $\frac{x^2}{2} - 8 = 0$
- d) $5x^2 = 2x$
The Quadratic Formula
For a complete equation $ax^2 + bx + c = 0$, use this formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$The $\pm$ gives two solutions: one by adding the square root and one by subtracting it. (The formula comes from rewriting the general equation by completing the square, so it works for every quadratic equation.)
If the number under the root is not a perfect square, leave the root in your answer or round it. For $x^2 - 4x + 1 = 0$: $x = \dfrac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \approx 3.73$ or $0.27$.
Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the following complete equations: a) $x^2 - 6x + 8 = 0$ b)…
Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the following complete equations:
- a) $x^2 - 6x + 8 = 0$
- b) $x^2 + x - 12 = 0$
Checking Solutions and Unknown Coefficients
A solution is a value of $x$ that makes the equation true. To test a value, substitute it and see if both sides match. You do not need to solve anything.
This also lets you find a missing coefficient: if you know a solution, substitute it and solve for the unknown.
Check, without solving the equation, if the proposed values of $x$ are a solution to the equality $x^2 - 4x + 3 = 0$: a) $x = 1$ b) $x = 0$ c) $x = 3$ d) $x = -1$
Check, without solving the equation, if the proposed values of $x$ are a solution to the equality $x^2 - 4x + 3 = 0$:
- a) $x = 1$
- b) $x = 0$
- c) $x = 3$
- d) $x = -1$
Given the quadratic equation $x^2 + kx + 12 = 0$, calculate the value of the coefficient $k$ knowing that one of the solutions to the equation is $x = 3$. Once $k$ is…
Given the quadratic equation $x^2 + kx + 12 = 0$, calculate the value of the coefficient $k$ knowing that one of the solutions to the equation is $x = 3$. Once $k$ is found, find the other solution to the equation.
Getting to Standard Form First
Many equations do not look quadratic at first. Expand, clear fractions, combine like terms, and move everything to one side. Two tools help:
To clear fractions, multiply every term of both sides by the LCM of the denominators.
Solve the following quadratic equation by first simplifying the algebraic expressions by expanding the algebraic identities: $$(2x - 3)^2 - (x + 1)^2 = -5$$ Remember…
Solve the following quadratic equation by first simplifying the algebraic expressions by expanding the algebraic identities:
$$(2x - 3)^2 - (x + 1)^2 = -5$$Remember that to apply the quadratic formula, you must first reduce the equation to the form $ax^2 + bx + c = 0$.
Solve the following equation by first calculating to simplify the sides and using algebraic identities where necessary:…
Solve the following equation by first calculating to simplify the sides and using algebraic identities where necessary:
$\frac{(2x - 3)^2}{8} - \frac{x(x + 1)}{2} = \frac{x - 5}{4} - 1$
The Discriminant: How Many Solutions?
The expression under the square root, $\Delta = b^2 - 4ac$, is called the discriminant. It tells you how many real solutions there are without solving.
| $\Delta = b^2 - 4ac$ | Real solutions |
|---|---|
| $\Delta > 0$ | Two different solutions |
| $\Delta = 0$ | One solution (a repeated root): $x = -\dfrac{b}{2a}$ |
| $\Delta < 0$ | No real solution (no real square root of a negative) |
The discriminant is also useful when an equation has an unknown parameter.
You can also ask for no real solutions. Then you need $\Delta < 0$, which gives an inequality for the unknown.
Calculate the value of the discriminant $\Delta = b^2 - 4ac$ for each equation and indicate whether it has two solutions, a single repeated root, or no real solution…
Calculate the value of the discriminant $\Delta = b^2 - 4ac$ for each equation and indicate whether it has two solutions, a single repeated root, or no real solution (you do not need to calculate the solutions):
- a) $x^2 - 4x + 4 = 0$
- b) $x^2 + 2x + 5 = 0$
- c) $x^2 - 5x - 1 = 0$
Study the discriminant ($\Delta = b^2 - 4ac$) of the following equations and determine, without solving them, how many real solutions each one has. Justify your answer…
Study the discriminant ($\Delta = b^2 - 4ac$) of the following equations and determine, without solving them, how many real solutions each one has. Justify your answer in each case:
- a) $x^2 - 6x + 9 = 0$
- b) $2x^2 + 3x + 5 = 0$
- c) $3x^2 - 4x - 2 = 0$
What value should the constant term take in equation a) so that it suddenly has no solution?
Given the quadratic equation $x^2 - (k-2)x + k = 0$, determine what values the parameter $k$ must take so that the equation has a single real solution (repeated root).
Given the quadratic equation $x^2 - (k-2)x + k = 0$, determine what values the parameter $k$ must take so that the equation has a single real solution (repeated root).
Equations Quadratic in Form
In $x^4 - 5x^2 + 4 = 0$ only $x^4$ and $x^2$ appear. Use a substitution: let $t = x^2$, so $x^4 = (x^2)^2 = t^2$. Some problems call these biquadratic equations.
Biquadratic equations can be solved by making a change of variable $t = x^2$. Use this technique to find the four real solutions of the following equation: $$x^4 - 13x^2…
Biquadratic equations can be solved by making a change of variable $t = x^2$. Use this technique to find the four real solutions of the following equation:
$$x^4 - 13x^2 + 36 = 0$$Equations with Fractions in $x$
When the unknown is in a denominator, multiply by the LCM of the denominators to clear it. This often produces a quadratic equation. But a denominator can never be $0$, so write down which values of $x$ are not allowed, and check your answers against that list.
The next example is a classic: work problems. If a job takes $x$ hours alone, you finish $\dfrac{1}{x}$ of the job each hour. The rates (jobs per hour) of two workers add.
Solve the following equation. To do this, find the least common multiple of the denominators to eliminate the unknowns from them, and check that the solutions obtained…
Solve the following equation. To do this, find the least common multiple of the denominators to eliminate the unknowns from them, and check that the solutions obtained do not make any denominator zero:
$\frac{x}{x - 2} + \frac{2}{x} = \frac{x^2}{x^2 - 2x}$
Two 3D printers working together take $2$ hours to complete a complex model. If they worked separately, the older printer would take $3$ hours more than the new printer.…
Two 3D printers working together take $2$ hours to complete a complex model. If they worked separately, the older printer would take $3$ hours more than the new printer. How long would each one take to make the model on its own? (Hint: remember that the work done in one hour is the reciprocal of the total time).
Number and Triangle Problems
In word problems, name the unknown, write the equation, solve, and then reject solutions that do not make sense: a negative length, a fractional number of people, and so on.
For a right triangle, use the Pythagorean Theorem: $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$. If the sides are three consecutive integers, call them $x$, $x + 1$, $x + 2$. The hypotenuse is the longest side, so it is $x + 2$:
$$x^2 + (x + 1)^2 = (x + 2)^2$$The product of two consecutive natural numbers is equal to their sum increased by $19$ units. What numbers are we talking about? Hint: Remember that if the first number…
The product of two consecutive natural numbers is equal to their sum increased by $19$ units. What numbers are we talking about?
Hint: Remember that if the first number is $x$, its consecutive is expressed as $x + 1$.
Calculate the dimensions of a right triangle knowing that its sides are three consecutive integers.
Calculate the dimensions of a right triangle knowing that its sides are three consecutive integers.
Area Problems
For rectangles, write each side with the unknown and use $\text{area} = \text{length} \times \text{width}$. A border of width $x$ around a rectangle adds $x$ on both sides, so each dimension grows by $2x$.
Dividing every term by a common factor (like the $2$ above) makes the numbers smaller and the formula easier.
A video game programmer is designing an interface for a new application. They need to create a rectangular button whose base is $5\text{ cm}$ longer than its height. If…
A video game programmer is designing an interface for a new application. They need to create a rectangular button whose base is $5\text{ cm}$ longer than its height. If the total area that the button must occupy on the screen is $84\text{ cm}^2$, what must the dimensions (base and height) of the button be?
Set up a quadratic equation and solve it.
We want to place a wooden frame of constant width $x$ around a rectangular photograph of $10\text{ cm}$ by $15\text{ cm}$. If we know that the total area (photo plus…
We want to place a wooden frame of constant width $x$ around a rectangular photograph of $10\text{ cm}$ by $15\text{ cm}$. If we know that the total area (photo plus frame) must be $300\text{ cm}^2$, what must the width $x$ of the frame be?
A landscaping design company wants to surround a rectangular swimming pool $8 \text{ m}$ long by $6 \text{ m}$ wide with an artificial grass border of uniform width $x$.…
A landscaping design company wants to surround a rectangular swimming pool $8 \text{ m}$ long by $6 \text{ m}$ wide with an artificial grass border of uniform width $x$. If the total area of the enclosure (pool plus grass border) must be exactly $80 \text{ m}^2$, calculate how much the width $x$ of the border must measure.
Problems with Two Unknowns
Sometimes a problem has two unknowns linked by two facts. Use the simpler fact (usually a sum or perimeter) to write one unknown in terms of the other, then substitute into the second fact.
We have a cable of length $40 \text{ cm}$ which is cut into two pieces to form two different squares. If the sum of the areas of the two squares is $58 \text{ cm}^2$,…
We have a cable of length $40 \text{ cm}$ which is cut into two pieces to form two different squares. If the sum of the areas of the two squares is $58 \text{ cm}^2$, how much does the side of each of these squares measure?
Height Models
A thrown or launched object follows a parabola. Its height $h$ at time $t$ is a quadratic expression in $t$. To find when it is at a certain height, set the formula equal to that height and solve.
A surveillance drone takes off and follows a trajectory whose height in feet with respect to time in seconds is given by the expression $h(t) = -2t^2 + 20t + 5$.…
A surveillance drone takes off and follows a trajectory whose height in feet with respect to time in seconds is given by the expression $h(t) = -2t^2 + 20t + 5$. Calculate at what instants of time the drone is exactly at a height of $53 \text{ meters}$.
Back to the start: For the video-game box, $x^2 = 144$ gives $x = \pm\sqrt{144} = \pm 12$. A side length cannot be negative, so each side of the base is exactly $12$ inches.