Solving Quadratic Equations: Quadratic Formula

The correct quadratic formula is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. It is used to find the solutions (roots) of a quadratic equation.

A negative discriminant means there are no real roots. If it is zero, there is one real root, and if it is positive, there are two distinct real roots.

The discriminant of a quadratic equation is $\Delta =b^2-4ac$. For $x^2+4x+4=0$, $a=1$, $b=4$, and $c=4$. The discriminant is $4^2-4(1)(4)=16-16=0$, which indicates one real root.

Using the quadratic formula with $a=1$, $b=-6$, and $c=8$, the discriminant is $(-6{)}^2-4(1)(8)=36-32=4$. The roots are $x=\frac{6\pm 2}{2}$, which gives $x=4$ and $x=2$.

Using the quadratic formula with $a=1$, $b=4$, and $c=-5$, the discriminant is $4^2-4(1)(-5)=16+20=36$. The roots are $x=\frac{-4\pm 6}{2}$, which gives $x=1$ and $x=-5$.

Using the quadratic formula with $a=4$, $b=-8$, and $c=3$, the discriminant is $(-8{)}^2-4(4)(3)=64-48=16$. The roots are $x=\frac{8\pm 4}{8}$, which gives $x=\frac{3}{2}$ and $x=\frac{1}{2}$.