Quadratic Equations
A quadratic equation can be solved two ways: factorisation when it splits neatly, or the quadratic formula when it doesn’t. The discriminant tells you which kind of roots to expect before you even solve it.
1. Standard Form and the Quadratic Formula
The quantity b² − 4ac is called the discriminant. It tells you the nature of the roots without solving the equation:
| Discriminant | Nature of Roots |
|---|---|
| > 0 | Two distinct real roots |
| = 0 | Two equal real roots |
| < 0 | No real roots |
Split the middle term: 2x² + 4x − 3x − 6 = 2x(x+2) − 3(x+2) = (x+2)(2x−3)
Setting each factor to zero: x = −2 or x = 3/2
a=1, b=−4, c=−1. Discriminant = (−4)² − 4(1)(−1) = 16 + 4 = 20
x = [4 ± √20] / 2 = 2 ± √5
For equal roots, discriminant = 0: k² − 4(1)(9) = 0 → k² = 36 → k = ±6
Let speed = x km/h. Time taken = 360/x. At speed (x+5), time = 360/(x+5), which is 1 hour less:
360/x − 360/(x+5) = 1 → 360(x+5) − 360x = x(x+5) → 1800 = x² + 5x → x² + 5x − 1800 = 0
Discriminant = 25 + 7200 = 7225 = 85². x = (−5+85)/2 = 40 (rejecting the negative root, since speed can’t be negative).
Speed = 40 km/h.
- Forgetting the ± sign in the quadratic formula, which gives only one of the two roots.
- In word problems, forgetting to reject a root that doesn’t make physical sense (e.g. a negative speed, length, or time).
- Sign errors when computing the discriminant, especially when c is negative — always substitute carefully with brackets.
Key Vocabulary Recap
Syllabus Reference
Practice questions
Quadratic Equations — Practice
Covers: factorisation, the quadratic formula, discriminant and nature of roots, and situational problems. 10 questions.
Solve x² − 5x + 6 = 0 by factorisation.
Solve 2x² + x − 6 = 0 by factorisation.
Solve x² − 4x − 1 = 0 using the quadratic formula.
Find the discriminant of 2x² − 3x + 5 = 0 and state the nature of its roots.
Find the value of k such that x² − kx + 9 = 0 has equal roots.
Solve 3x² − 2x − 1 = 0 using the quadratic formula.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the journey. Find the speed of the train.
For what values of k does 9x² + kx + 1 = 0 have two equal real roots?
Solve x² + 7x + 10 = 0 by factorisation.
The sum of the squares of two consecutive positive integers is 313. Find the integers.
More Practice
Solve for x: x² − (√2+1)x + √2 = 0 by factorisation.
Check whether x = −2 is a root of 3x² + 13x + 14 = 0.
Find two consecutive odd positive integers, the sum of whose squares is 290.
The product of two consecutive positive integers is 306. Find the integers.
Find the roots of the equation 1/x − 1/(x−2) = 3, where x ≠ 0, 2.
A person on tour has ₹360 for expenses. If he extends his tour by 4 days, he must cut his daily expenses by ₹3. Find the original duration of the tour.
Find k so that the equation (k+1)x² − 2(k+1)x + 1 = 0 has equal roots (k ≠ −1).
Two water taps together can fill a tank in 9⅜ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time each tap takes.
If the equation (1+m²)x² + 2mcx + (c²−a²) = 0 has equal roots, prove that c² = a²(1+m²).
Solve for real roots: x² − 3√5x + 10 = 0.
Worked solutions
Show worked solutions
Quadratic Equations — Practice: Solutions
x²−5x+6=0.
2x²+x−6=0.
x²−4x−1=0 by formula.
Discriminant of 2x²−3x+5=0.
x²−kx+9=0 has equal roots. Find k.
3x²−2x−1=0 by formula.
Train 360km, speed+5 → time−1hr.
9x²+kx+1=0 equal roots. Find k.
x²+7x+10=0.
Sum of squares of consecutive integers = 313.
More Practice — Solutions
x²−(√2+1)x+√2=0.
Is x=−2 a root of 3x²+13x+14=0?
Consecutive odd integers, sum of squares=290.
Consecutive integers, product=306.
1/x − 1/(x−2) = 3.
₹360 budget, extend 4 days, ₹3 less per day.
(k+1)x²−2(k+1)x+1=0 equal roots.
Two taps, together 9⅜ hrs, larger 10hrs less than smaller.
Prove c²=a²(1+m²) given equal roots.
x²−3√5x+10=0.
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