The Quadratic Formula

The standard form for a quadratic equation (a parabola) is ax2+bx+c. To find the roots of the equation (where it crosses the x-axis), one approach is to use the quadratic formula. This should be memorized!

x=-b±b2-4ac2a

Example: 2x2+11x-21=0

In this problem, a = 2, b = 11 and c = -21

x=-11±112-4∙2∙-212∙2
=-11±121+1684
=-11±2894
=-11±174

So: x=-11+174=64=32
Or: x=-11-174=-284=-7

This means that 2x2+11x-21=x-32x+7
=2x-3x+7

Where did that 2x-3 come from?!

Remember, we are solving this equation, looking for where the graph crosses the x-axis. That means that we can set each set of parentheses equal to zero. The x+7 part doesn’t need any changes, but we prefer not to have a fraction in the factored form. So we can set x-32=0 and play with it.

x-32=0
2x-32=20
2x-3=0

The graph will cross the x-axis at 3/2 and -7, with the vertex at

-2.75, -36.125.

The Quadratic Formula with Irrational Roots.

The quadratic formula comes in very handy when an equation cannot be factored in the normal way because it crosses the x-axis at an irrational number (like 32).

Example: 5x2+3x-6=0
x=-3±32-4∙5∙-62∙5
=-3±9+12010
=-3±12910
So 5x2+3x-6=0 factors to x--3+12910x--3-12910=0.

This is approximately equal to x-0.84x+1.43=0. Clearly, trying to factor this the regular way would be a nightmare!

The Quadratic Formula with Imaginary Roots

Sometimes the graph never crosses the x-axis at all (it has imaginary roots like 2+3i)

Example: -3x2-3x-5

x=--3±-32-4-3-52-3
=3±9-60-6
=3±-51-6
=3±i51-6

So x=3+i51-6 or x=3-i51-6 and the vertex is at approximately -0.5, -4.

The Discriminant:

The discriminant is the part of the quadratic formula that is under the square root bracket.

x=-b±b2-4ac2a

If b2-4ac is a positive number, the parabola will have two real roots (it will cross the x-axis twice).

If b2-4ac is a negative number, the parabola will have no real roots (it will never cross the x-axis).

If b2-4ac=0, the parabola will have one real root (it will “nick” the x-axis at the vertex).

Note: If the discriminant is a perfect square, you know that the equation is factorable.

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