ICSE Class 10
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๐Ÿ“– Summaries โ€บ Mathematics

Quadratic Equations (in one variable)

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Chapter in a nutshell: A quadratic equation has the form $ax^2+bx+c=0\ (a\ne0)$. Solve it by factorisation, completing the square, or the quadratic formula. The discriminant $b^2-4ac$ tells the nature of the roots, and the roots relate to the coefficients by sum $=-b/a$, product $=c/a$.

1. Key Ideas

  • Standard form: $ax^2+bx+c=0$, with $a\ne0$ (a, b, c real).
  • A quadratic has at most two roots (solutions).
  • Three solving methods: factorisation, completing the square, quadratic formula.

2. Formulas (quick reference)

  • Quadratic formula: $\;x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
  • Discriminant: $\;D=b^2-4ac$
  • Nature of roots:
- $D>0$ โ†’ real and distinct roots - $D=0$ โ†’ real and equal roots $\left(x=\dfrac{-b}{2a}\right)$ - $D<0$ โ†’ no real roots (imaginary)
  • Sum of roots $\alpha+\beta=-\dfrac{b}{a}$; Product of roots $\alpha\beta=\dfrac{c}{a}$.
  • Forming an equation from roots: $x^2-(\alpha+\beta)x+\alpha\beta=0$.

3. Methods (steps)

  • Factorisation: write $ax^2+bx+c$ as a product of two factors (split the middle term so the two parts multiply to $ac$ and add to $b$), set each factor = 0.
  • Completing the square: make a perfect square $(x+\frac{b}{2a})^2$ and solve.
  • Formula: substitute a, b, c into the quadratic formula.

4. Prerequisite Formulas (earlier classes)

  • Factorisation by splitting the middle term; identities $(a\pm b)^2=a^2\pm2ab+b^2$, $a^2-b^2=(a+b)(a-b)$.
  • Surds: simplifying $\sqrt{\ }$ (for the formula's root part).

5. Worked Example (one, for the method)

Q. Solve $x^2-5x+6=0$. Solution: Split: $x^2-2x-3x+6=0\Rightarrow x(x-2)-3(x-2)=0\Rightarrow(x-2)(x-3)=0$. So x = 2 or 3. (Check: sum 5 = โˆ’b/a, product 6 = c/a.)

6. Common Mistakes to Avoid

  • Forgetting $a\ne0$ (else it isn't quadratic).
  • Sign error in $-b$ in the formula or in the discriminant.
  • Saying $D<0$ has equal roots โ€” it has no real roots.
  • Splitting the middle term so the parts don't multiply to ac.
  • Mixing up sum $=-b/a$ and product $=c/a$.

7. Likely Exam Questions (with crisp answers)

  1. Write the standard form of a quadratic. โ†’ $ax^2+bx+c=0,\ a\ne0$.
  2. State the quadratic formula. โ†’ $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
  3. What is the discriminant? โ†’ $D=b^2-4ac$.
  4. When are the roots real and equal? โ†’ When $D=0$.
  5. When are there no real roots? โ†’ When $D<0$.
  6. Sum and product of roots? โ†’ $-b/a$ and $c/a$.
  7. Form a quadratic with roots 2 and 3. โ†’ $x^2-5x+6=0$.
  8. How many roots does a quadratic have? โ†’ At most two.
  9. Name the three methods of solving. โ†’ Factorisation, completing the square, formula.
  10. Find D for $x^2-4x+4=0$ and the nature of roots. โ†’ D = 0 โ†’ real and equal.

Formulas โ€” Current & Prerequisite

Current

  • Standard form $ax^2+bx+c=0,\ a\ne0$.
  • Quadratic formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.
  • Discriminant $D=b^2-4ac$: $D>0$ real & distinct, $D=0$ real & equal, $D<0$ no real roots.
  • Sum of roots $\alpha+\beta=-\dfrac{b}{a}$; product $\alpha\beta=\dfrac{c}{a}$.

Prerequisite

  • Factorisation (splitting the middle term); $\sqrt{ }$ surds; identities $(a\pm b)^2=a^2\pm2ab+b^2$, $a^2-b^2=(a-b)(a+b)$.
  • Solving linear equations; framing equations from word problems.