Quadratic Functions that are always positive or negative

In this post, we’ll look at quadratic functions that are always positive or negative.

We can find out whether a quadratic function is always positive or negative by completing the square.

To do so, we’ll first rewrite a quadratic function of the form ax² + bx +c into a(x+g)² + h by completing the square.

All square numbers are positive. Hence, (x+g)² will always be greater than or equal to 0 or (x+g)²≥ 0.

When a is positive, i.e. a> 0, a(x+g)²≥ 0. Then a(x+g)² + h≥ h for all values of x. This means that a(x+g)² + h has a minimum value of h. If h is positive, then a(x+g)² + h is always positive.

When a is negative, i.e. a< 0, a(x+g)²≤ 0. Then a(x+g)² + h≤ h for all values of x. This means that a(x+g)² + h has a maximum value of h. If h is negative, then a(x+g)² + h is always negative.

Summary of Determining whether quadratic function is always positive or negative

To determine whether a quadratic function of the form ax² + bx +c, we can complete the square and rewrite it into a(x+g)² + h.

If a > 0 AND h > 0, a(x+g)² + h > 0 for all values of x. Minimum value is h. Since h >0, quadratic function is always positive.
If a <0 AND h < 0, a(x+g)² + h < 0 for all values of x. Maximum value is h. Since h < 0, quadratic function is always negative.

Note: Quadratic functions that do not follow the above 2 criteria are sometimes positive and sometimes negative.

Questions

For the quadratic functions below, determine which ones are (i) always positive (ii) always negative (iiii) take both negative and positive values.

(a) x² -3x +5

(b) -x² +4x -5

Solutions to questions

(a) Completing the square for x² -3x +5 gives (x-1.5)² +2.75.

(x-1.5)² +2.75 is in the form of a(x+g)² + h, a = 1 >0, and h =2.75 > 0.

Minimum value of x² -3x +5 or (x-1.5)² +2.75 is 2.75.

Hence, (x-1.5)² +2.75 > 0 for all values of x. Hence, x² -3x +5 is always positive.

(b) Completing the square for -x² +4x -5 gives -(x-2)² -1.

-(x-2)² -1 is in the form of a(x+g)² + h, a = -1 <0, and h =-1 < 0.

Maximum value of -x² +4x -5 or -(x-2)² -1 is -1.

Hence, -(x-2)² -1 < 0 for all values of x. Hence, -x² +4x -5 is always negative.

(x-0.5)² – 4.25 is in the form of a(x+g)² + h, a = 1 >0, and h =-4.25 < 0.

Minimum value of x²-x-4 or (x-0.5)² – 4.25 is -4.25.

Hence, (x-0.5)² – 4.25 ≥ – 4.25 for all values of x. Hence, x²-x-4 is sometimes positive and sometimes negative.

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