calculus online - Free exercises and solutions to help you succeed!

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:

$$ax^2+bx+c=a(x-x_1)(x-x_2)$$

Press here for exercises and solutions in quadratic equations

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