Kit Library / Mathematics / Algebra
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
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.
💡 Key Idea
The essential definition you must know.
A quadratic equation in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \neq 0$. The term $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term.
↳ A quadratic equation is a second-degree polynomial equation set to zero.
📖 Definition
The universal format for all quadratics.
Any quadratic equation can be rewritten in this standard form by moving all terms to one side and simplifying. This form is the starting point for solving by factorization, completing the square, or using the quadratic formula.
↳ Always rewrite a quadratic equation in the form $ax^2 + bx + c = 0$ before solving.
✏️ Example
Spot a, b, and c quickly.
For the equation $2x^2 - 3x + 5 = 0$, the coefficients are:
↳ Always note the signs: $b$ and $c$ include their signs.
💡 Key Idea
The values that satisfy the equation.
A root (or solution) of a quadratic equation $ax^2 + bx + c = 0$ is a real number $\alpha$ such that $a\alpha^2 + b\alpha + c = 0$. In other words, substituting the root for $x$ makes the equation true. A quadratic equation can have at most two distinct roots.
↳ Roots are the x-values where the quadratic expression equals zero.
⚙️ Process
Step-by-step method to find roots.
↳ Factorization converts a quadratic into two simple linear equations.
✏️ Example
See the method in action.
Solve $x^2 - 5x + 6 = 0$ by factorization.
↳ For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$.
⚠️ Common Mistake
A classic error that costs marks.
When solving an equation like $x^2 = 3x$, students often divide both sides by $x$ to get $x = 3$, losing the root $x = 0$. Always bring all terms to one side first: $x^2 - 3x = 0$, then factor: $x(x - 3) = 0$, giving $x = 0$ or $x = 3$.
↳ Never divide by a variable that could be zero; factor instead.
➗ Formula
A universal solver for any quadratic.
This formula gives the roots directly, even when factorization is not easy. The expression under the square root, $b^2 - 4ac$, is called the discriminant.
↳ The quadratic formula works for every quadratic equation.
⭐ Important Fact
Predict roots without solving.
For $ax^2 + bx + c = 0$, the discriminant $D = b^2 - 4ac$ determines the nature of the roots:
↳ The discriminant tells you how many real roots exist.
✏️ Example
Apply the formula step by step.
Solve $2x^2 - 4x - 6 = 0$ using the quadratic formula.
↳ Substitute carefully and simplify the fraction.
⚖️ Comparison
Which method to choose?
↳ Use factorization for simple quadratics; otherwise, use the formula.
🌍 Real World
Real-life applications of quadratics.
Quadratic equations model projectile motion (height of a thrown object over time), area problems (finding dimensions of a rectangle given area), and profit maximization in business. For example, the height $h(t)$ of a ball thrown upward is often $h(t) = -4.9t^2 + v_0 t + h_0$.
↳ Quadratics describe many natural and economic phenomena.
🎯 Exam Tip
Verify your answers quickly.
After solving, substitute each root back into the original equation to ensure it gives $0$. This catches arithmetic errors and confirms you didn't miss a root.
↳ Always verify roots by substitution.
🧠 Memory Trick
A catchy tune to memorize.
Sing the quadratic formula to the tune of *"Pop Goes the Weasel"*: *"x equals negative b, plus or minus the square root, of b-squared minus 4ac, all over 2a."*
↳ Music helps memorize the formula.
❓ Did You Know
Quadratics are thousands of years old.
Babylonian mathematicians solved quadratic-type problems as early as 2000 BC, using geometric methods. The general solution we use today was formalized much later.
↳ Quadratics have a long history in mathematics.
Quadratic equations are second-degree polynomial equations of the form $ax^2 + bx + c = 0$. They can be solved by factorization or the quadratic formula, and the discriminant determines the nature of the roots. This kit…
Full explanation, worked examples & memory tricks in the app
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. Which of the following is the standard form of a quadratic equation?
BeginnerD · $ax^2 + bx + c = 0$
The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a \neq 0$.
Q2. In the equation $2x^2 - 3x + 5 = 0$, what is the value of $a$?
BeginnerC · 2
In the standard form $ax^2 + bx + c = 0$, $a$ is the coefficient of $x^2$, which is 2.
Q3. Which of the following is a quadratic equation?
BeginnerA · $x^2 + 2x + 1 = 0$
A quadratic equation has degree 2, so $x^2 + 2x + 1 = 0$ is quadratic.
Q4. The roots of the equation $x^2 - 9 = 0$ are:
BeginnerC · 3 and -3
$x^2 - 9 = (x-3)(x+3) = 0$, so $x = 3$ or $x = -3$.
Q5. Which of the following is NOT a quadratic equation?
BeginnerD · $x^3 + x^2 + 1 = 0$
A quadratic equation has degree 2; $x^3 + x^2 + 1 = 0$ is cubic.
Q6. The equation $x^2 + 2x + 1 = 0$ has two distinct real roots.
BeginnerB · False
The discriminant is $4 - 4 = 0$, so it has equal real roots, not distinct.
Q7. The roots of the equation $x^2 - 5x + 6 = 0$ are 2 and 3.
BeginnerA · True
Factorizing gives $(x-2)(x-3)=0$, so roots are indeed 2 and 3.
Q8. What are the roots of the equation $x^2 - 5x + 6 = 0$?
IntermediateD · 2 and 3
Factorizing: $x^2 - 5x + 6 = (x-2)(x-3) = 0$, so $x=2$ or $x=3$.
Q9. Which of the following is a root of $2x^2 - 4x = 0$?
IntermediateC · 0
Factorizing: $2x(x-2) = 0$, so $x=0$ or $x=2$. Thus 0 is a root.
Q10. The equation $x^2 + 4x + 4 = 0$ has roots:
IntermediateB · -2 and -2
$x^2 + 4x + 4 = (x+2)^2 = 0$, so $x = -2$ (double root).
Q11. What is the product of the roots of $x^2 - 7x + 12 = 0$?
IntermediateD · 12
For $ax^2 + bx + c = 0$, product of roots = $c/a = 12/1 = 12$.
Q12. The sum of the roots of $3x^2 - 6x + 2 = 0$ is:
IntermediateA · 2
Sum of roots = $-b/a = -(-6)/3 = 2$.
Q13. Which of the following equations has roots 5 and -2?
IntermediateD · $x^2 - 3x - 10 = 0$
Sum = 3, product = -10, so equation is $x^2 - 3x - 10 = 0$.
Q14. The factorization of $x^2 - 4x - 5$ is:
IntermediateC · $(x-5)(x+1)$
We need two numbers whose product is -5 and sum is -4: -5 and 1, so $(x-5)(x+1)$.
Q15. What is the discriminant of $x^2 + 2x + 1 = 0$?
IntermediateB · 0
Discriminant $D = b^2 - 4ac = 4 - 4(1)(1) = 0$.
Q16. The equation $x^2 - 4x + 4 = 0$ has:
IntermediateA · Two equal real roots
Discriminant $D = 16 - 16 = 0$, so roots are real and equal.
Q17. The roots of $x^2 + 6x + 9 = 0$ are:
IntermediateD · -3 and -3
$x^2 + 6x + 9 = (x+3)^2 = 0$, so $x = -3$ (double root).
Q18. The value of $k$ for which $x = 2$ is a root of $x^2 - 3x + k = 0$ is:
IntermediateA · 2
Substitute $x=2$: $4 - 6 + k = 0 \Rightarrow k = 2$.
17 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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Word Match
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True False
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Memory Match
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Flashcard Battle
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Speed Quiz
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Sequence Builder
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Revision Battle
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