The quadratic formula solves ax² + bx + c = 0 where a b and c are numbers and a ≠ 0. The solution is x = [-b ± √(b² – 4ac)] / (2a). The ± means to calculate both + and – versions giving two possible answers.
Quadratic Equations deals with polynomials having highest power of x as 2 in the standard form ax² + bx + c = 0 where a b and c are real numbers and a ≠ 0. To find solutions or zeros we use multiple methods like factorization square completion and quadratic formula. The solutions x = [-b ± √(b² – 4ac)] / (2a) help us solve real-world problems about areas speed time and projectile motion in CBSE Class 10 Mathematics.
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Given equation: 3x² – 2x – 1 = 0
Using quadratic formula:
x = [-b ± √(b² – 4ac)]/2a
Here:
a = 3
b = -2
c = -1
Substituting:
x = [2 ± √(4 – 4(3)(-1))]/6
x = [2 ± √(4 + 12)]/6
x = [2 ± √16]/6
x = [2 ± 4]/6
For + sign:
x = (2 + 4)/6
x = 6/6
x = 1
For – sign:
x = (2 – 4)/6
x = -2/6
x = -1/3
Therefore roots are: 1 and -1/3
To verify:
3(1)² – 2(1) – 1 = 0
3(-1/3)² – 2(-1/3) – 1 = 0
Hence, 1, -1/3 are the correct roots.
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