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Quadratic Equations

Introduction

A quadratic equation is an equation that can be written in the standard form $a x^{2} + b x + c = 0$ , where $a \neq 0$ . The highest power of the variable is two, so a quadratic equation is a second-degree polynomial equation. In Grades 9–10, quadratic equations connect algebraic manipulation, quadratic functions, graphs, and mathematical modelling.

You will learn to recognize quadratic equations, solve them by several methods, check solutions, interpret graphs, use the discriminant, and choose a method that fits the structure of a problem. You will also apply quadratics to situations involving area, height, motion, and optimization.

Learning goals: By the end of this aiMOOC, you should be able to explain what a quadratic equation is, solve quadratic equations by factoring, square roots, completing the square, and the quadratic formula, connect solutions to x-intercepts of a parabola, interpret the discriminant, verify solutions by substitution, and model suitable real-world situations.

The graph above shows the basic quadratic function $f (x) = x^{2}$ . Its U-shaped graph is called a parabola. Quadratic equations and parabolas are closely connected because solving $f (x) = 0$ means finding where the graph meets the x-axis.

What Makes an Equation Quadratic?

A quadratic equation contains a nonzero $x^{2}$ term and no power of $x$ higher than two. The standard form is:

$a x^{2} + b x + c = 0, a \neq 0$

Here, $a$ , $b$ , and $c$ are coefficients. The coefficient $a$ controls the quadratic term, $b$ controls the linear term, and $c$ is the constant term.

For example, $2 x^{2} - 5 x - 3 = 0$ is quadratic because the highest exponent is two and the coefficient of $x^{2}$ is not zero. The equation $4 x - 7 = 0$ is linear, not quadratic, while $x^{3} + x = 0$ is cubic.

Before solving, rearrange the equation so that one side is zero. For example:

$x^{2} + 7 = 5 x$

becomes

$x^{2} - 5 x + 7 = 0$ .

Equations and Functions

The expression $a x^{2} + b x + c$ can define a quadratic function $f (x) = a x^{2} + b x + c$ . The equation $a x^{2} + b x + c = 0$ asks for the inputs where $f (x) = 0$ . These inputs are called solutions, roots, or zeros.

In the graph above, the roots are the x-values where the parabola crosses the x-axis. This graphical meaning helps you check whether an algebraic answer is reasonable.

This Khan Academy video demonstrates solving a quadratic equation by factoring. As you watch, notice how the zero-product property turns a product equal to zero into separate linear equations.

Solving by Factoring

Factoring is often the fastest method when the quadratic expression can be written as a product of linear factors. The key rule is the zero-product property: if $p q = 0$ , then $p = 0$ or $q = 0$ .

Example:

$x^{2} - 5 x + 6 = 0$

Factor the left-hand side:

$(x - 2) (x - 3) = 0$

Set each factor equal to zero:

$x - 2 = 0$ or $x - 3 = 0$

Therefore, $x = 2$ or $x = 3$ .

You can check both answers in the original equation. For $x = 2$ , $2^{2} - 5 (2) + 6 = 0$ . For $x = 3$ , $3^{2} - 5 (3) + 6 = 0$ . Both are valid.

Factoring When the Leading Coefficient Is Not One

Consider $2 x^{2} + 7 x + 3 = 0$ . One useful strategy is to search for two binomials whose product gives the quadratic:

$(2 x + 1) (x + 3) = 0$

Then:

$2 x + 1 = 0$ or $x + 3 = 0$

so the solutions are $x = - \frac{1}{2}$ and $x = - 3$ .

Factoring is efficient when the factors are easy to recognize. If the expression does not factor conveniently over the integers, another method may be better.

Solving by Taking Square Roots

If a quadratic has the form $(x - h)^{2} = k$ , isolate the squared expression and take both square roots:

$x - h = \pm \sqrt{k}$ .

Example:

$(x - 4)^{2} = 25$

gives

$x - 4 = \pm 5$ ,

so $x = 9$ or $x = - 1$ .

The symbol $\pm$ matters because both a positive and a negative number can have the same positive square. Forgetting the negative square root is a common error.

If $k < 0$ , there is no real number whose square equals $k$ . In Grades 9–10, you can usually report that the equation has no real solutions unless your course has introduced complex numbers.

Completing the Square

Completing the square rewrites a quadratic expression so that part of it becomes a perfect-square trinomial. This method is important because it works even when factoring is difficult, reveals the vertex form of a quadratic function, and leads directly to the quadratic formula.

For $x^{2} + 6 x + 5 = 0$ :

$x^{2} + 6 x = - 5$

Half of 6 is 3, and $3^{2} = 9$ . Add 9 to both sides:

$x^{2} + 6 x + 9 = 4$

Then:

$(x + 3)^{2} = 4$

Take square roots:

$x + 3 = \pm 2$

so $x = - 1$ or $x = - 5$ .

The image gives a geometric way to think about completing the square: pieces representing the quadratic and linear terms can be rearranged so that adding a smaller square creates a complete square.

Use this video to compare the algebraic steps with the geometric idea. Pay attention to the rule that whatever you add to one side of an equation must also be added to the other side.

Completing the Square When the Leading Coefficient Is Not One

For an equation such as $2 x^{2} + 8 x - 10 = 0$ , first divide every term by 2:

$x^{2} + 4 x - 5 = 0$

Then complete the square:

$x^{2} + 4 x = 5$

$x^{2} + 4 x + 4 = 9$

$(x + 2)^{2} = 9$

so $x + 2 = \pm 3$ , giving $x = 1$ or $x = - 5$ .

This example shows why simplifying first can make the method easier.

The Quadratic Formula

Every quadratic equation $a x^{2} + b x + c = 0$ with $a \neq 0$ can be solved using the quadratic formula:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

To use the formula accurately:

Standard form: Rearrange the equation to $a x^{2} + b x + c = 0$ .
Coefficient: Identify $a$ , $b$ , and $c$ , including their signs.
Substitution: Substitute the values into the formula.
Square root: Evaluate the expression under the square root.
Simplification: Simplify both values produced by $\pm$ .

Example: Solve $2 x^{2} - 3 x - 2 = 0$ .

Here $a = 2$ , $b = - 3$ , and $c = - 2$ . Substitute:

$x = \frac{- (- 3) \pm \sqrt{(- 3)^{2} - 4 (2) (- 2)}}{2 (2)}$

$x = \frac{3 \pm \sqrt{9 + 16}}{4}$

$x = \frac{3 \pm 5}{4}$

Therefore, $x = 2$ or $x = - \frac{1}{2}$ .

The image connects substitution in the quadratic formula with the x-intercepts of a parabola.

This worked-example video is useful for checking sign handling, substitution, and simplification.

Where the Formula Comes From

The quadratic formula can be derived from the general equation $a x^{2} + b x + c = 0$ by completing the square. You do not need to memorize every derivation step immediately, but understanding the derivation helps you see that the formula is not an isolated trick.

As you study the derivation, look for the same operations you use when solving a specific equation: move terms, divide by the leading coefficient, complete a square, take square roots, and isolate the variable.

The Discriminant

The expression under the square root in the quadratic formula is called the discriminant:

$D = b^{2} - 4 a c$ .

The discriminant tells you how many real solutions a quadratic equation has before you calculate the solutions.

Positive discriminant: If $D > 0$ , there are two distinct real solutions.
Zero discriminant: If $D = 0$ , there is one repeated real solution.
Negative discriminant: If $D < 0$ , there are no real solutions.

The three parabolas illustrate the three cases. Two x-axis crossings correspond to two real roots, one touching point corresponds to one repeated real root, and no x-axis crossing corresponds to no real roots.

Use the discriminant as a prediction tool. It can help you decide what kind of answer to expect before you finish a calculation.

Graphs, Roots, and the Vertex

A quadratic function $f (x) = a x^{2} + b x + c$ has a parabolic graph. Important features include the vertex, axis of symmetry, x-intercepts, and y-intercept.

The x-coordinate of the vertex is:

$x = - \frac{b}{2 a}$ .

Substitute this x-value into the function to find the y-coordinate. The vertical line through the vertex is the axis of symmetry. If $a > 0$ , the parabola opens upward and the vertex is a minimum. If $a < 0$ , the parabola opens downward and the vertex is a maximum.

For a quadratic equation, the x-intercepts are especially important because they are the real solutions. A graph can provide exact solutions when intercepts are clearly known, but most hand-drawn or calculator graphs give approximate solutions. Algebraic methods are usually better when an exact answer is required.

Choosing a Solving Method

Good problem solving includes choosing an efficient method rather than applying the same method every time.

Structure of the equation	Useful method	Reason
A product of factors equals zero	Factoring	The zero-product property gives the roots quickly
A squared expression equals a number	Taking square roots	The square can be undone directly
A monic quadratic is close to a perfect square	Completing the square	The equation can be rewritten in square form
Any quadratic in standard form	Quadratic formula	The formula works for every quadratic equation
A visual estimate or interpretation is needed	Graphing	Intercepts show approximate real roots

A strong solver can use more than one method and compare them. For example, $x^{2} - 9 = 0$ can be factored as $(x - 3) (x + 3) = 0$ or solved by taking square roots from $x^{2} = 9$ . Both methods give $x = \pm 3$ .

Checking Solutions and Common Errors

Checking is part of solving. Substitute each proposed solution into the original equation. If both sides are equal, the value is a solution.

Common errors include losing the negative sign when identifying $b$ , forgetting the $\pm$ symbol, dividing only some terms by a number, factoring incorrectly, taking the square root of a negative number while working only with real numbers, and rounding too early.

A useful habit is to estimate first. If a graph suggests roots near 2 and 5, an algebraic answer of 200 should make you recheck your work.

Real-World Models

Quadratic equations appear when a quantity depends on the square of a variable. Examples include area models, projectile height under simplified assumptions, revenue or profit models, and optimization problems.

Suppose a rectangular garden has width $x$ metres and length $x + 5$ metres, with area 84 square metres. Then:

$x (x + 5) = 84$

$x^{2} + 5 x - 84 = 0$

$(x + 12) (x - 7) = 0$

The algebraic solutions are $x = - 12$ and $x = 7$ , but a physical width cannot be negative. The meaningful solution is therefore $x = 7$ metres, giving a length of 12 metres.

This example shows an important modelling principle: an algebraic solution may be mathematically correct but not meaningful in the context of the problem.

A Height Model

A simplified height model might be $h (t) = - 5 t^{2} + 20 t + 1$ , where $t$ is time in seconds and $h$ is height in metres. Solving $- 5 t^{2} + 20 t + 1 = 0$ estimates when the object reaches ground level. The vertex of the parabola gives the maximum height and the time at which it occurs.

When using a model, always state units and interpret which solutions make sense. Negative time values, for example, may not belong to the situation being modelled.

Interactive Tasks

Quiz: Test Your Knowledge

Which equation is quadratic? (3x squared plus 2x minus 5 equals zero) (!7x minus 4 equals zero) (!x cubed plus x equals zero) (!5 divided by x equals two)

What condition must the leading coefficient satisfy in a quadratic equation? (It must not be zero) (!It must equal one) (!It must be positive) (!It must be an integer)

What are the solutions of x squared minus 9 equals zero? (x equals 3 or x equals negative 3) (!x equals 9 only) (!x equals 3 only) (!x equals negative 9 only)

Which method is especially efficient when a quadratic is already written as a product equal to zero? (Factoring) (!Graph translation) (!Long division) (!Elimination)

What does a positive discriminant indicate? (Two distinct real solutions) (!One repeated real solution) (!No real solutions) (!Infinitely many solutions)

What does a discriminant of zero indicate? (One repeated real solution) (!Two distinct real solutions) (!No real solutions) (!The equation is linear)

What is the graph of a quadratic function called? (Parabola) (!Circle) (!Hyperbola) (!Straight line)

Where do real roots appear on the graph of a quadratic function? (At the x intercepts) (!At every y intercept) (!Only at the vertex) (!Only on the axis of symmetry)

Why should you substitute a proposed root into the original equation? (To check that it satisfies the equation) (!To change the equation into a function) (!To remove the quadratic term) (!To create an extra root)

Which method works for every quadratic equation in standard form? (Quadratic formula) (!Factoring over integers) (!Taking square roots immediately) (!Guessing from a table)

Memory Game

Coefficient	A number multiplying a variable term
Root	A value that makes the quadratic expression equal zero
Vertex	The turning point of a parabola
Discriminant	The expression that predicts the number of real solutions
Factorization	Rewriting an expression as a product
Parabola	The graph of a quadratic function
Substitution	Replacing a variable with a chosen value to check or evaluate

Drag and Drop

Match the correct terms.	Topic
Factoring	Best when the expression splits conveniently into linear factors
Square-root method	Best when a squared expression is isolated
Completing the square	Rewrites a quadratic into a perfect-square form
Quadratic formula	General method that works for every quadratic equation
Graphing	Useful for estimating roots from x-axis crossings

Match each solving method with the description that best explains when or why to use it.

Crossword Puzzle

Parabola	What is the U-shaped graph of a quadratic function called?
Discriminant	What expression tells you the number of real roots?
Factorization	What process rewrites a polynomial as a product?
Vertex	What is the turning point of a parabola called?
Intercepts	What points show where a graph crosses an axis?
Symmetry	What property makes the two sides of a parabola mirror each other?

LearningApps

Cloze Text

A quadratic equation can be written in standard form with a nonzero

. The graph of the related quadratic function is a

. Values that make the quadratic expression equal zero are called

. Factoring uses the

when a product equals zero. Completing the square creates a

within the equation. The general method that works for every quadratic is the

. The expression under its square root is the

. A positive discriminant predicts

. A zero discriminant predicts

. Substituting a proposed answer into the original equation helps you

.

Open-Ended Tasks

Easy

Quadratic Equation Poster: Create a one-page poster that shows standard form, labels the coefficients, and includes one correctly solved example.
Root Check Cards: Make four study cards with quadratic equations on the front and verified solutions on the back.
Parabola Sketch: Draw a quadratic graph with a labelled vertex, axis of symmetry, x-intercepts, and y-intercept, then explain each feature in two or three sentences.
One-Minute Quadratic Video: Record a short video explaining why the symbol plus-or-minus is necessary when solving a squared equation.

Standard

Method Comparison: Solve the same quadratic equation by two different methods and write a comparison of efficiency, clarity, and risk of error.
Error Analysis: Invent three realistic mistakes in quadratic-equation work, exchange them with a partner, and write corrected solutions with explanations.
Quadratics Interview: Interview a teacher, engineer, technician, designer, or other professional about where they meet parabolic shapes, optimization, or quadratic models, then summarize what you learned.
Projectile Data Project: Use a safe tossed-ball video or teacher-provided data to estimate height over time, fit or use a quadratic model, and explain what the roots and vertex mean in context.

Advanced

Quadratic Formula Derivation: Produce a step-by-step derivation of the quadratic formula by completing the square and annotate why each transformation preserves equivalence.
Discriminant Investigation: Create several quadratic equations with positive, zero, and negative discriminants, graph them, and explain the connection between algebraic predictions and x-axis intersections.
Optimization Design Challenge: Design a realistic maximum-area or minimum-cost problem that leads to a quadratic function, solve it, and justify why the vertex gives the optimum.
Quadratic Modelling Report: Collect or generate data that approximately follows a parabola, fit a quadratic model with a suitable digital tool, evaluate its limitations, and present your findings as a report or video.

Learning Assessment

Method Selection Assessment: Given six quadratic equations in different forms, choose an efficient solving method for each and justify every choice before solving.
Multiple Representation Assessment: For one quadratic, show the equation in standard form, a factored or vertex form, its graph, and its solutions, then explain how the representations correspond.
Discriminant Reasoning Assessment: Predict the number of real roots of several quadratics from their discriminants, then verify the predictions algebraically or graphically.
Modelling Assessment: Build and solve a quadratic equation from a realistic area, height, or revenue situation, and explain which algebraic solutions are meaningful in context.
Error Diagnosis Assessment: Analyze a worked solution containing a sign or factoring error, identify the first incorrect step, repair the solution, and explain why the correction works.
Transfer Assessment: Create a new quadratic problem whose solution cannot be read off immediately, solve it with two methods, and compare the exact and graphical results.

Evidence of Learning

Knowledge: You can explain standard form, roots, factors, the vertex, the axis of symmetry, the discriminant, and the relationship between a quadratic equation and its graph.
Skills: You can rearrange equations, factor suitable quadratics, take square roots correctly, complete the square, use the quadratic formula, interpret graphs, and verify answers.
Products: Your portfolio can include solved examples, annotated graphs, an error analysis, a modelling task, a poster, a short explanatory video, or a digital investigation.
Reasoning: You can choose a solving method based on structure, predict the number of real solutions, explain why a method works, and identify unreasonable results.
Transfer: You can create or analyze a real-world model, interpret roots and the vertex in context, use units correctly, and reject algebraically valid answers that are not meaningful for the situation.

OERs on the Topic

The English Wikipedia article below provides an open reference for definitions, solution methods, graphs, the quadratic formula, and further mathematical context.

You can also explore related open topics through Quadratic function, Parabola, Polynomial, Factoring, Completing the square, Quadratic formula, and Discriminant.

Linked Learning Areas

Quadratic Equations

Quadratic equations connect several learning areas. In Algebra, you transform expressions and solve equations. In Functions, you connect symbolic rules with graphs. In Geometry, you study the structure and symmetry of parabolas. In Physics, simplified motion models can produce quadratic relationships. In data analysis, quadratic models can approximate curved trends. In Mathematical modelling, you decide whether algebraic solutions make sense in a real situation.

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