Chapter 4 · Question 4

State the quadratic formula and the condition for real roots.

Answer

For

ax^2+bx+c=0

,

a\ne0

, the roots are

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

. Real roots exist when

b^2-4ac\ge0

.

Use

x=(-b\pm\sqrt{D})/(2a)

, where

D=b^2-4ac

. Real roots need

D\ge0

.

The discriminant is

D=b^2-4ac

. The roots of

ax^2+bx+c=0

are

x=\frac{-b+\sqrt D}{2a}

and

x=\frac{-b-\sqrt D}{2a}

. If

D>0

, roots are real and distinct; if

D=0

, roots are real and equal; if

D<0

, there are no real roots.

For $ax^2+bx+c=0$ , $a\ne0$ , the roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .