📖 Definition
What Makes an Equation Quadratic?
Degree 2 is the whole game.
↳ If a = 0, the equation collapses to linear — so a ≠ 0 is the defining condition.
Kit Library / Mathematics / Algebra
⚡ Topic Learning KitQuadratic Equations
English 19 leveled MCQs 25 flashcards 9 games Free
Shared by a Veda learner · Generated with Veda AI
Veda Bites
📖 Definition
➗ Formula
💡 Key Idea
⚙️ Process
⚙️ Process
⚠️ Common Mistake
⚠️ Common Mistake
💡 Key Idea
🌍 Real World
💡 Visualization
🧠 Memory Trick
🎯 Exam Tip
💡 Shortcut
⭐ Important Fact
🛠️ Application
The whole idea, one bite at a time
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
📖 Definition
➗ Formula
The Quadratic Formula: Your Universal Solver
One formula solves every quadratic.
↳ Memorise this formula — it is the fallback when factorisation fails.
💡 Key Idea
The Discriminant Tells You the Nature of Roots
One number predicts your roots.
↳ Compute D first — it tells you what kind of answer to expect before solving.
⚙️ Process
Solving by Factorisation: Split the Middle Term
Split b, crack the equation open.
↳ Factorisation is quick but only works cleanly when roots are rational.
⚙️ Process
Completing the Square: Rewrite as a Perfect Square
Turn any quadratic into (x + p)².
↳ Completing the square is the derivation behind the quadratic formula — and the basis for vertex form.
⚠️ Common Mistake
Forgetting the ± When Taking Square Roots
Missing ± costs you half the marks.
↳ Every square root step in a quadratic produces two branches — never drop the negative.
⚠️ Common Mistake
Sign Errors in the Quadratic Formula
One wrong sign ruins both roots.
↳ Substitute into a bracketed template — it prevents the most common sign slips.
💡 Key Idea
Sum and Product of Roots (Vieta's Relations)
Roots hide inside the coefficients.
↳ You can often answer root-related questions without ever solving the equation.
🌍 Real World
Where Quadratics Show Up in Real Life
Parabolas are everywhere you look.
↳ Any situation with a squared term and a peak or dip is a quadratic in disguise.
The Graph Is a Parabola
Roots are where the curve hits zero.
The discriminant D tells you how many times the parabola crosses the x-axis: twice, once, or never.
↳ Sketching the parabola makes the number and location of roots visually obvious.
🧠 Memory Trick
Remember the Formula with a Rhythm
Sing it once, never forget it.
Chant: "minus b, plus or minus, root of b-squared minus four-a-c, all over two-a." Repeating it aloud three times locks the structure in memory.
↳ Rhythm beats rote — say the formula, don't just read it.
🎯 Exam Tip
Exam Strategy: Pick the Fastest Method
Right method saves minutes.
↳ Choosing the right method is worth more marks than solving faster.
Quick Check: Verify Your Roots
Two seconds to catch errors.
After finding roots α and β, quickly verify that α + β = −b/a and αβ = c/a. If either fails, you have made an arithmetic slip.
↳ A 5-second Vieta check catches most calculation mistakes before you submit.
⭐ Important Fact
Every Quadratic Has Exactly Two Roots
Two roots — always, counting repeats.
By the Fundamental Theorem of Algebra, a quadratic has exactly two roots over the complex numbers. They may be real and distinct, real and equal, or complex conjugates — but there are always two.
↳ If you find only one root, either it is a repeated root or you missed the second.
🛠️ Application
Building a Quadratic from Its Roots
Reverse the process — roots to equation.
↳ Sum and product of roots is a two-way street — use it to build or to check.
Smart notes
1
What you'll study, topic by topic
Quadratic Equations
18 minQuadratic equations are polynomial equations of degree two, written in the standard form ax² + bx + c = 0 where a ≠ 0. They can be solved by factorisation, completing the square, or the quadratic formula, and their roots…
- A quadratic equation has the standard form $ax^2 + bx + c = 0$ where $a \neq 0$.
- The condition $a \neq 0$ is essential — otherwise the equation becomes linear.
- The quadratic formula is $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Full explanation, worked examples & memory tricks in the app
Leveled MCQ practice
Try the smart MCQs from this kit
All 19 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Warm-up · 8 questions
Q1. Which of the following is the standard form of a quadratic equation?
Beginner
A
$a^2x + b = 0$
B
$ax + b = 0$
C
$ax^2 + bx + c = 0$
D
$ax^3 + bx^2 + cx + d = 0$
Show the answer
C · $ax^2 + bx + c = 0$
A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \neq 0$. The highest power of the variable must be 2.
Q2. In the equation $ax^2 + bx + c = 0$, which condition must hold for it to be quadratic?
Beginner
A
$a = 0$
B
$c \neq 0$
C
$b = 0$
D
$a \neq 0$
Show the answer
D · $a \neq 0$
If $a = 0$, the $x^2$ term vanishes and the equation becomes linear. The condition $a \neq 0$ is essential.
Q3. What is the discriminant of a quadratic equation $ax^2 + bx + c = 0$?
Beginner
A
$4ac - b^2$
B
$b^2 + 4ac$
C
$\sqrt{b^2 - 4ac}$
D
$b^2 - 4ac$
Show the answer
D · $b^2 - 4ac$
The discriminant is $D = b^2 - 4ac$. It is the expression under the square root in the quadratic formula.
Q4. If the discriminant $D > 0$, what can you say about the roots?
Beginner
A
No real roots
B
Two distinct real roots
C
Exactly one real root
D
Two equal real roots
Show the answer
B · Two distinct real roots
When $D > 0$, the square root is a positive real number, giving two different real values for $x$.
Q5. If the discriminant $D = 0$, what is the nature of the roots?
Beginner
A
No real roots
B
Two equal real roots
C
Three real roots
D
Two distinct real roots
Show the answer
B · Two equal real roots
When $D = 0$, the $\pm$ part adds and subtracts zero, so both roots are the same: $x = -b/(2a)$.
Q6. If the discriminant $D < 0$, what is the nature of the roots?
Beginner
A
One real and one complex root
B
Two equal real roots
C
Two distinct real roots
D
No real roots
Show the answer
D · No real roots
When $D < 0$, the square root of a negative number is not real, so the equation has no real roots (the roots are complex conjugates).
Q7. If $a > 0$ in $y = ax^2 + bx + c$, which way does the parabola open?
Beginner
A
Downward
B
Rightward
C
Leftward
D
Upward
Show the answer
D · Upward
The sign of $a$ determines the direction. If $a > 0$, the parabola opens upward (like a U). If $a < 0$, it opens downward.
Q8. The quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves every quadratic equation with $a \neq 0$.
Beginner
A
True
B
False
Show the answer
A · True
The quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves every quadratic equation with $a \neq 0$.
Building up · 6 questions
Q9. What is the sum of the roots $\alpha$ and $\beta$ of $ax^2 + bx + c = 0$?
Intermediate
A
$\dfrac{c}{a}$
B
$-\dfrac{b}{a}$
C
$-\dfrac{c}{a}$
D
$\dfrac{b}{a}$
Show the answer
B · $-\dfrac{b}{a}$
The sum of the roots is $\alpha + \beta = -\dfrac{b}{a}$. This comes from expanding $(x - \alpha)(x - \beta) = 0$.
Q10. What is the product of the roots $\alpha$ and $\beta$ of $ax^2 + bx + c = 0$?
Intermediate
A
$\dfrac{b}{a}$
B
$-\dfrac{c}{a}$
C
$\dfrac{c}{a}$
D
$-\dfrac{b}{a}$
Show the answer
C · $\dfrac{c}{a}$
The product of the roots is $\alpha\beta = \dfrac{c}{a}$. This is a standard result from comparing coefficients.
Q11. A ball is thrown upward and its height in metres after $t$ seconds is given by $h = -5t^2 + 20t + 1$. What type of equation models this?
Intermediate
A
Exponential equation
B
Quadratic equation
C
Linear equation
D
Cubic equation
Show the answer
B · Quadratic equation
The height depends on $t^2$, making it a quadratic equation. Projectile motion is a classic real-world application of quadratics.
Q12. For the quadratic $y = ax^2 + bx + c$, at what $x$-value is the vertex located?
Intermediate
A
$x = -\dfrac{b}{a}$
B
$x = \dfrac{c}{a}$
C
$x = -\dfrac{b}{2a}$
D
$x = \dfrac{b}{2a}$
Show the answer
C · $x = -\dfrac{b}{2a}$
The vertex (turning point) of the parabola is at $x = -\dfrac{b}{2a}$, which is also the axis of symmetry.
Q13. Which method is fastest to solve $x^2 - 5x + 6 = 0$ if the roots are simple integers?
Intermediate
A
Graphing
B
Factorisation
C
Quadratic formula
D
Completing the square
Show the answer
B · Factorisation
Factorisation is fastest when roots are simple integers. Here $(x-2)(x-3)=0$ gives roots 2 and 3 immediately.
Q14. The graph of a quadratic equation $y = ax^2 + bx + c$ is a parabola. What do the roots of the equation represent on this graph?
Intermediate
A
The $y$-intercepts
B
The $x$-intercepts
C
The axis of symmetry
D
The vertex
Show the answer
B · The $x$-intercepts
The roots are the values of $x$ where $y = 0$, i.e., where the parabola crosses the $x$-axis. These are the $x$-intercepts.
Topper level · 5 questions
Q15. A student solves $2x^2 + 3x - 5 = 0$ using the quadratic formula and writes only $x = \frac{-3 + \sqrt{49}}{4} = 1$. What mistake did the student make?
Advanced
A
Simplified the square root incorrectly
B
Forgot to include the $\pm$ and missed the second root
C
Used the wrong formula
D
Substituted $a$, $b$, $c$ incorrectly
Show the answer
B · Forgot to include the $\pm$ and missed the second root
The quadratic formula has a $\pm$, meaning there are two roots. The student only computed the $+$ case and missed $x = \frac{-3 - 7}{4} = -\frac{5}{2}$.
Q16. For a quadratic equation, if the sum of roots is 5 and the product of roots is 6, what is the equation?
Advanced
A
$x^2 - 5x - 6 = 0$
B
$x^2 - 5x + 6 = 0$
C
$x^2 + 5x + 6 = 0$
D
$x^2 + 5x - 6 = 0$
Show the answer
B · $x^2 - 5x + 6 = 0$
If roots are $\alpha$ and $\beta$, the equation is $x^2 - (\alpha+\beta)x + \alpha\beta = 0$. Substituting sum $=5$ and product $=6$ gives $x^2 - 5x + 6 = 0$.
Q17. Given the quadratic equation \(2x^2 - 4x + k = 0\), if the roots are equal, what is the value of \(k\)?
Advanced
A
8
B
2
C
0
D
4
Show the answer
A · 8
The roots are equal when the discriminant \(D = b^2 - 4ac = 0\). Here, \(D = (-4)^2 - 4(2)(k) = 16 - 8k\). Setting this to zero gives \(16 - 8k = 0\) or \(k = 2\).
Q18. For the quadratic equation \(x^2 - 6x + 9 = 0\), what is the sum of the roots and how does it relate to the coefficients?
Advanced
A
9
B
6
C
3
D
0
Show the answer
B · 6
The sum of the roots for a quadratic \(ax^2 + bx + c = 0\) is given by \(- rac{b}{a}\). Here, \(b = -6\) and \(a = 1\), thus the sum of the roots is \(- rac{-6}{1} = 6$.
Q19. Given the quadratic equation \(3x^2 + 2x + c = 0\) has roots \(r_1\) and \(r_2\) such that \(r_1 + r_2 = - rac{2}{3}\) and \(r_1 r_2 = rac{c}{3}\), what is the value of \(c\) if the product of the roots is 4?
Advanced
A
10
B
12
C
8
D
6
Show the answer
B · 12
The product of the roots is given by \(r_1 r_2 = rac{c}{3}\). If \(r_1 r_2 = 4\), then \(4 = rac{c}{3}\) leads to \(c = 12\).
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