Quadratic Formula – Quadratic Equation

When given a quadratic equation:

ax2+bx+c=0,a0ax^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:

x1,2=b±b24ac2ax_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Usually the following is defined:

Δ=b24ac\Delta=b^2-4ac

Then it’s easy to see from the formula that the equation has 2 roots when

Δ>0\Delta>0

That means the graph crosses the x-axis twice.

The equation has one root when

Δ=0\Delta=0

Which means the graph crosses the x-axis exactly once.

And it has no real roots when

Δ<0\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:

x1+x2=bax_1+x_2=-\frac{b}{a}

x1x2=cax_1\cdot x_2=\frac{c}{a}

Also, with the roots you can factor the quadratic equation as follows:

ax2+bx+c=a(xx1)(xx2)ax^2+bx+c=a(x-x_1)(x-x_2)

Press here for exercises and solutions in quadratic equations

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