When given a quadratic equation:
ax^2+bx+c=0, a\neq 0
The points where the graph intersects the x-axis are called roots or zeros or solutions, which can be easily found with this quadratic formula:
x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
Usually the following is defined:
\Delta=b^2-4ac
Then it’s easy to see from the formula that the equation has 2 roots when
\Delta>0
That means the graph crosses the x-axis twice.
The equation has one root when
\Delta=0
Which means the graph crosses the x-axis exactly once.
And it has no real roots when
\Delta<0
And that means the graph doesn’t cross the x-axis at all.
In addition, the roots of the equation also follow these formulas:
x_1+x_2=-\frac{b}{a}
x_1\cdot x_2=\frac{c}{a}
Also, with the roots you can factor the quadratic equation as follows:
ax^2+bx+c=a(x-x_1)(x-x_2)
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