Quadratic Equations
Your school wants to paint a square mural that covers exactly $144$ square feet of the gym wall. How long should each side be? You need a number $x$ so that $x \cdot x = 144$. This is a new kind of equation: the unknown is squared. These are called quadratic equations, and by the end of this page you will be able to solve any of them, even the ones where you can't just "see" the answer.
Standard Form
In a linear equation, $x$ appears only to the first power. In a quadratic equation, the highest power of $x$ is $2$.
Why $a \neq 0$? If $a = 0$, the $x^2$ term disappears and you are left with a linear equation.
To find $a$, $b$ and $c$, first move every term to one side so the other side is $0$.
The graph of $y = ax^2 + bx + c$ is a parabola. The solutions of $ax^2 + bx + c = 0$ are the $x$-values where the parabola crosses the $x$-axis. Move the sliders. Can you make the parabola stop touching the $x$-axis?
Identify which of the following equations are quadratic. For those that are, write them in their general form ($ax^2 + bx + c = 0$) and state the value of their…
Identify which of the following equations are quadratic. For those that are, write them in their general form ($ax^2 + bx + c = 0$) and state the value of their coefficients $a$, $b$ and $c$:
- a) $3x^2 - 5 = x + 2$
- b) $x^2 + 3x - x^2 = 10$
- c) $5x - 3 = 0$
- d) $(x + 2) \cdot (x - 2) = 0$
Incomplete Equations: The Square Root Method
When the $x$ term is missing ($b = 0$), you don't need a formula. Isolate $x^2$ and take the square root, just like in the mural problem.
Why two signs? Because both $12^2$ and $(-12)^2$ equal $144$.
Calculate the solution to these incomplete quadratic equations of the type $ax^2 + c = 0$ (remember that they can have two solutions, one or none): a) $x^2 - 49 = 0$ b)…
Calculate the solution to these incomplete quadratic equations of the type $ax^2 + c = 0$ (remember that they can have two solutions, one or none):
- a) $x^2 - 49 = 0$
- b) $2x^2 - 32 = 0$
- c) $x^2 + 25 = 0$
- d) $5x^2 = 0$
We want to design a square rug for a room. If we know that the total area it must occupy is $144$ $m^2$, how long must the side of the rug be? Write the corresponding…
We want to design a square rug for a room. If we know that the total area it must occupy is $144$ $m^2$, how long must the side of the rug be? Write the corresponding quadratic equation and solve it.
A town council wants to install a square advertising banner on the facade of a sports center to announce an eSports event. If the total surface area occupied by the…
A town council wants to install a square advertising banner on the facade of a sports center to announce an eSports event. If the total surface area occupied by the banner is $144 m^2$, set up a quadratic equation and calculate the side length of this banner.
Incomplete Equations: Factoring Out $x$
When the constant term is missing ($c = 0$), every term has an $x$ in it. Factor it out and use this idea:
It works because multiplying two nonzero numbers never gives $0$. So for $ax^2 + bx = 0$:
$$x^2 - 7x = 0 \ \Rightarrow\ x(x - 7) = 0 \ \Rightarrow\ x = 0 \ \text{ or } \ x - 7 = 0 \ \Rightarrow\ x = 0 \text{ or } x = 7$$Some problems call this step extracting a common factor. It is the same as factoring out $x$.
Solve the following incomplete quadratic equations by extracting a common factor. Do not forget that one of the solutions is always $x = 0$: a) $x^2 - 7x = 0$ b)…
Solve the following incomplete quadratic equations by extracting a common factor. Do not forget that one of the solutions is always $x = 0$:
- a) $x^2 - 7x = 0$
- b) $3x^2 + 12x = 0$
- c) $x^2 = 5x$
- d) $4x^2 - 2x = 0$
Solve the following incomplete quadratic equations using the most appropriate method (extracting a common factor or direct square root): a) $3x^2 - 27 = 0$ b)…
Solve the following incomplete quadratic equations using the most appropriate method (extracting a common factor or direct square root):
- a) $3x^2 - 27 = 0$
- b) $x^2 + 5x = 0$
- c) $2x^2 = 8x$
Building an Equation from Its Solutions
The Zero Product Property also works backwards. If the solutions are $r_1$ and $r_2$, then $(x - r_1)(x - r_2) = 0$ is an equation with exactly those solutions.
The parabola crosses the $x$-axis at exactly $2$ and $7$.
Emma and Ethan are playing "guess the equation". Emma says: "My equation has the solutions $x_1 = 3$ and $x_2 = -5$ and the coefficient of the $x^2$ term is $1$". Write…
Emma and Ethan are playing "guess the equation". Emma says: "My equation has the solutions $x_1 = 3$ and $x_2 = -5$ and the coefficient of the $x^2$ term is $1$".
Write Emma's equation in its general form ($x^2 + bx + c = 0$). Hint: Use the factorized form $(x - x_1) \cdot (x - x_2) = 0$.
The Quadratic Formula
For a complete equation (with $b$ and $c$ both nonzero), there is a formula that always works:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$The $\pm$ gives two answers: one by adding the square root and one by subtracting it. (The formula comes from completing the square on $ax^2 + bx + c = 0$, but you only need to know how to use it.)
Sometimes the square root is not a whole number. Then use a calculator and round. For $x^2 - 4x + 1 = 0$: $b^2 - 4ac = 16 - 4 = 12$, so
$$x = \frac{4 \pm \sqrt{12}}{2} \approx \frac{4 \pm 3.46}{2} \quad\Rightarrow\quad x \approx 3.73 \ \text{ or } \ x \approx 0.27$$Solve the following complete quadratic equations using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ a) $x^2 - 5x + 6 = 0$ b) $x^2 + 2x - 8 = 0$ c)…
Solve the following complete quadratic equations using the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$- a) $x^2 - 5x + 6 = 0$
- b) $x^2 + 2x - 8 = 0$
- c) $x^2 - 10x + 25 = 0$
A garden designer wants to surround a circular fountain with a flower border. The equation relating the measurements of the design is $x(x - 8) = -15$, where $x$ is a…
A garden designer wants to surround a circular fountain with a flower border. The equation relating the measurements of the design is $x(x - 8) = -15$, where $x$ is a measurement in feet. Solve the equation using the quadratic formula to find the possible values of $x$.
Simplify First, Then Solve
Many equations don't start in standard form. They may have parentheses, a squared binomial, or fractions. Clean them up first.
The picture shows why $(x + 2)^2$ is not $x^2 + 4$: you also get two rectangles of area $2x$.
Simplify the following expression to the general form $ax^2 + bx + c = 0$ and find its solutions by applying the formula: $$(x + 2)^2 - 7 = 2x + 1$$
Simplify the following expression to the general form $ax^2 + bx + c = 0$ and find its solutions by applying the formula:
$$(x + 2)^2 - 7 = 2x + 1$$A video game programmer is designing the formula to calculate a character's critical experience. After simplifying the code, the resulting equation is the following:…
A video game programmer is designing the formula to calculate a character's critical experience. After simplifying the code, the resulting equation is the following:
$$\frac{(x + 2)^2}{3} - \frac{x(x - 1)}{2} = \frac{x^2 + 7}{6}$$Expand and simplify to the general form $ax^2 + bx + c = 0$ and find the values of $x$ that make the formula work.
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. If $\Delta$ is positive you can take its square root and get $\pm$ two different answers. If $\Delta = 0$, then $\pm 0$ gives the same answer twice. If $\Delta$ is negative, there is no real square root.
| $\Delta = b^2 - 4ac$ | Solutions |
|---|---|
| $\Delta > 0$ | Two different real solutions |
| $\Delta = 0$ | One solution (a repeated root) |
| $\Delta < 0$ | No real solutions |
Without solving the equations, calculate the value of the discriminant ($\Delta = b^2 - 4ac$) and indicate whether they have two solutions, one repeated root or no real…
Without solving the equations, calculate the value of the discriminant ($\Delta = b^2 - 4ac$) and indicate whether they have two solutions, one repeated root or no real solutions:
- a) $x^2 - 6x + 9 = 0$
- b) $x^2 - 5x + 4 = 0$
- c) $2x^2 + 3x + 5 = 0$
In the mathematics laboratory, we are analysing the behavior of different projectiles. Without solving the following equations, use the discriminant (…
In the mathematics laboratory, we are analysing the behavior of different projectiles. Without solving the following equations, use the discriminant ($\Delta = b^2 - 4ac$) to indicate how many real solutions each one has:
- a) $x^2 - 6x + 9 = 0$
- b) $2x^2 + 3x + 5 = 0$
- c) $x^2 - 5x + 4 = 0$
A parabolic ramp is going to be built for a new skatepark. For the ramp to be safe and have a single point of support on the ground, the discriminant of the equation…
A parabolic ramp is going to be built for a new skatepark. For the ramp to be safe and have a single point of support on the ground, the discriminant of the equation describing its trajectory, $x^2 - 10x + k = 0$, must be equal to zero.
What value must the coefficient $k$ have for the ramp to have a single solution (repeated root)? Once $k$ is found, solve the equation to find the point of support.
Word Problems: Areas
Quadratic equations show up whenever two lengths multiply. Use the same plan every time:
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Define the unknown ("Let $x$ = the width in feet").
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Write every other quantity in terms of $x$.
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Set up the equation (draw a picture first).
-
Solve it.
-
Throw out answers that make no sense (a negative length) and check the answer in the story.
When the answer is not a whole number, use the formula, then round the result as the problem asks (for example, to two decimal places).
Find the dimensions of the rectangle in the figure knowing that its area is $15$ $cm^2$. To do this, set up a quadratic equation and solve it using the quadratic formula.
Find the dimensions of the rectangle in the figure knowing that its area is $15$ $cm^2$. To do this, set up a quadratic equation and solve it using the quadratic formula.
The base of a new tablet is rectangular. We know that its base is $4 cm$ longer than its height and that its total surface area is $60 cm^2$. Calculate the dimensions…
The base of a new tablet is rectangular. We know that its base is $4 cm$ longer than its height and that its total surface area is $60 cm^2$. Calculate the dimensions (base and height) of the tablet by solving the corresponding quadratic equation.
An urban artist wants to paint a square mural on a wall. However, they decide to enlarge the original design by increasing its side by $2$ feet. By doing so, the total…
An urban artist wants to paint a square mural on a wall. However, they decide to enlarge the original design by increasing its side by $2$ feet. By doing so, the total area of the mural triples compared to the original design.
Calculate how long the side of the mural was originally (round the result to two decimal places if necessary).
Word Problems: Pythagorean Theorem and Open Boxes
Two classic geometry setups give quadratic equations.
Right triangles. The Pythagorean Theorem says $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (some problems write Pythagoras' theorem). If the legs are expressions in $x$, squaring them produces an $x^2$ equation.
Open boxes. Cut a square of side $x$ out of each corner of a sheet, fold up the sides, and the base shrinks by $2x$ in each direction (you remove $x$ from both ends).
An electric scooter $130$ in. long has been left leaning against a wall. The distance from the foot of the wall to the base of the scooter is $70$ in. less than the…
An electric scooter $130$ in. long has been left leaning against a wall. The distance from the foot of the wall to the base of the scooter is $70$ in. less than the height the scooter reaches on the wall.
If we call the height on the wall $x$, apply Pythagoras' theorem to set up a quadratic equation and calculate this height.
A rectangular sheet of cardboard has a base of $20$ in. and a height of $15$ in. We want to cut out a square of side $x$ from each of its corners to make an open box. If…
A rectangular sheet of cardboard has a base of $20$ in. and a height of $15$ in. We want to cut out a square of side $x$ from each of its corners to make an open box. If we know that the base area of the resulting box must be $150$ square inches, calculate how long the side $x$ of the cut-out squares must be.
Word Problems: Number Puzzles
To solve "find the number" problems, first translate the words into algebra.
| Words | Algebra |
|---|---|
| a number | $n$ |
| the square of a number | $n^2$ |
| two consecutive integers | $n$ and $n + 1$ |
| the product of two numbers | $n(n + 1)$ |
| a number minus $3$, squared | $(n - 3)^2$ |
| four times the smaller one | $4n$ |
The product of an integer and the next consecutive integer is equal to $30$. Translate this statement into a quadratic equation and find what numbers they could be…
The product of an integer and the next consecutive integer is equal to $30$. Translate this statement into a quadratic equation and find what numbers they could be (remember that there are two possible pairs of numbers).
Find an integer such that if you add its own square to it, the result obtained is $42$. Hint: Remember to move all terms to one side to equate the equation to zero.
Find an integer such that if you add its own square to it, the result obtained is $42$. Hint: Remember to move all terms to one side to equate the equation to zero.
The product of two consecutive integers is equal to four times the smaller one plus $10$. Set up the corresponding quadratic equation and find what these two numbers…
The product of two consecutive integers is equal to four times the smaller one plus $10$. Set up the corresponding quadratic equation and find what these two numbers are. Is there more than one pair of numbers that satisfies the condition?
Word Problems: Real-World Models
Sometimes a formula gives a quantity that depends on a squared variable, like the height of something thrown into the air. To find when it reaches a certain value, set the formula equal to that value and solve.
A small toy drone is launched vertically upwards from the ground. The height $h$ (in feet) of the drone after a time $t$ (in seconds) is given by the expression: $$h =…
A small toy drone is launched vertically upwards from the ground. The height $h$ (in feet) of the drone after a time $t$ (in seconds) is given by the expression:
$$h = 30t - 5t^2$$At what times will the drone be exactly at a height of $25$ feet? Explain why you obtain two different solutions.
On a photography social network, the number of "Likes" on an influencer*'s post today has been such that, if $10$ is subtracted from it and the result is squared, the…
On a photography social network, the number of "Likes" on an influencer's post today has been such that, if $10$ is subtracted from it and the result is squared, the same value is obtained as multiplying the number of "Likes" by $5$ and adding $50$.
Calculate how many "Likes" this post has received today.
Back to the start: For the mural, $x^2 = 144$ gives $x = \pm\sqrt{144} = \pm 12$. A length can't be negative, so each side is exactly $12$ ft.