Question

3x² - 12x +9=0 using quadratic formula

Answer 1

The quadratic equation 3x² - 12x + 9 = 0 has two real roots when solved:

x₁ = 1 and x₂ = 3

Explanation:

✍ An equation of type ax² + bx + c = 0, can be solved, for example, using the quadratic formula:

`\boldsymbol{x=\frac{-b\pm\sqrt{b^(2)-4ac } }{2a} }`

either

`\boldsymbol{x=(-b\pm√(\Delta) )/(2a) }`

where

`\bf{\Delta=b^(2)-4ac }`

Identify the coefficients

a = 3, b = -12 and c = 9

Calculate the discriminant value

Δ = b² - 4ac

Δ = (-12)² - 4.3.9 = 144 - 12.9

Δ = 144 - 108 = 36

Enter the values of a, b and the discriminant value in the quadratic formula

`\boldsymbol{x=(-b\pm√(\Delta ) )/(2a) }`

`\boldsymbol{x=(-(-12\pm√(36) )/(2\cdot3 ) }`

`\boldsymbol{x=((12\pm√(36) )/(6 ) \ ==== > \ (Solution \ general) }`

As we can see above, the discriminant (Δ) of this equation is positive (Δ> 0) which means that there are two real roots (two solutions), x₁ and x₂.

`\boldsymbol{x_(1)=(-12-√(36) )/(6)=(12-6)/(6)=(6)/(6)=1 \ === > (First \ solution) }`

To find x₁, just choose the positive sign before the square root. Later,

`\boldsymbol{x_(1)=(12+√(36) )/(6)=(12+6)/(6)=(18)/(6)=3 \ === > (Second \ solution) }`