How to solve quadratic equations by using the quadratic formula: formula, proof, examples, and their solutions.
For any quadratic equation (a ≠ 0),
x can be found by using the quadratic formula.
Prove the quadratic formula
by completing the square.
Move +c to the right side.
Divide both sides by a.
Change +(b/a)x to +2⋅x⋅[b/(2a)].
Write +[b/(2a)]2 on both sides.
Factor the left side by using the formula:
x2 + 2⋅x⋅[b/(2a)] + [b/(2a)]2 = (x + [b/(2a)])2.
Change -(c/a) to -[(c⋅4a)/(a⋅4a)].
Then add it with +b2/4a2.
Square root both sides.
Move +b/(2a) to the right side.
Then x = [(-b ± √b2 - 4ac)]/(2a).
Put a = 1, b = 3, c = -2 into the formula.
Put a = 4, b = -1, c = +5 into the formula.
√-79 is not a real number
because the number inside the radical sign
cannot be (-).So this equation has no real roots.
If you know about complex numbers,
there's a way to find the roots.
Complex roots of a quadratic equation