# 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

(c) x² -x – 4

## 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.

(c) Completing the square for x²-x-4 gives (x-0.5)² – 4.25.

(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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