Unit 39: Zeros in Quadratics

Zeros in Quadratics and Other Polynomials

One of the most crucial skills in solving quadratic problems is understanding the language used.  While in real life these are slightly different terms, when referring to a single quadratic, these words all mean the same thing:

Zeros = Roots = Solutions = Real Solutions = x-intercepts

So when a question asks you about the zeros of a quadratic, they are asking about the x-intercepts.  

The first graph has no intercepts, and thus no real solutions.  The second is tangent to the x-axis, so it has one solution at = 2.  The third graph has solutions at x = 1 and x = 3.

Factors

The first graph above shows an equation that would not factor.

The second graph shows an equation where (x – 2) MUST be a factor, because y = 0 when = 2.  

The third graph shows an equation where (-1) and (x – 3) MUST be factors.

If given a problem like:

An equation has roots at -1 and 5.  Which if the following must be a factor?

A) (x+5)
B) (x)
C) (x-1)
D) (x+1)

Strategy:

  • Rebuild the factored equation by forming binomials with the given roots (solutions).

If an equation has roots of -1 and 5, then in its factored form it would look like ((x+1)(x-5)). This is because those binomials have solutions of -1 and 5. Thus, the answer would be D.

Zeros in Other Equations

The exact same concept applies if you are asked about zeros in other equations.  For instance, given the graph below:

You can easily see that it has 4 x-axis intercepts, and thus would have 4 zeros or solutions.

Note that the approach above for identifying factors holds true regardless of the number of roots / solutions.  If given a polynomial like: 

(ax^3+bx^2+cx+d=0) with roots of -4, -2, and 3, 

you would apply exactly the same logic. 

Traps

The halfway vertex

If you know two zeros of a quadratic, then you actually also know the x value of the vertex. The vertex is always halfway between two zeros/roots/solutions/x-intercepts.