Practice Zeros of a Quadratic in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The zeros (or roots) of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c are the values of xx where f(x)=0f(x) = 0. Graphically, they are the xx-intercepts of the parabola.

The zeros are where the parabola crosses or touches the xx-axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

Showing a random 20 of 50 problems.

Example 1

hard
Find the zeros of f(x)=x2+4x+1f(x) = x^2 + 4x + 1 by completing the square.

Example 2

easy
How many real zeros does f(x)=(xโˆ’5)2f(x) = (x - 5)^2 have?

Example 3

medium
Find the zeros of f(x)=โˆ’x2+5x+6f(x) = -x^2 + 5x + 6.

Example 4

challenge
A quadratic f(x)=x2+bx+cf(x)=x^2+bx+c has zeros 22 and โˆ’5-5. Find bb and cc.

Example 5

easy
Find the zeros of f(x)=x2โˆ’100f(x) = x^2 - 100.

Example 6

hard
If a quadratic with leading coefficient 11 has zeros โˆ’1-1 and 44, write it in standard form.

Example 7

challenge
If f(x)=x2+px+qf(x) = x^2 + px + q has zeros rr and ss with r+s=5r + s = 5 and r2+s2=13r^2 + s^2 = 13, find pp and qq.

Example 8

medium
Find the sum and product of the zeros of f(x)=2x2+5xโˆ’3f(x) = 2x^2 + 5x - 3.

Example 9

easy
How many real zeros does f(x)=x2+25f(x) = x^2 + 25 have?

Example 10

easy
Find the zeros of h(x)=x2โˆ’1h(x) = x^2 - 1.

Example 11

easy
Find the zeros of f(x)=x2โˆ’5x+6f(x) = x^2 - 5x + 6.

Example 12

easy
Find the zeros of f(x)=x2โˆ’7x+10f(x) = x^2 - 7x + 10.

Example 13

easy
Find the zeros of f(x)=(xโˆ’3)(x+8)f(x) = (x - 3)(x + 8).

Example 14

easy
Is f(0)f(0) a zero of f(x)=x2โˆ’4f(x) = x^2 - 4?

Example 15

medium
If โˆ’4-4 is a zero of f(x)=x2+bx+4f(x) = x^2 + bx + 4, find bb.

Example 16

medium
If 33 is a zero of f(x)=x2+bxโˆ’12f(x) = x^2 + bx - 12, find bb.

Example 17

medium
Find the zeros of f(x)=2x2+5x+2f(x) = 2x^2 + 5x + 2.

Example 18

medium
Find the zeros of f(x)=3x2โˆ’7x+2f(x) = 3x^2 - 7x + 2.

Example 19

medium
Find the zeros of f(x)=x2โˆ’2xโˆ’8f(x) = x^2 - 2x - 8 by factoring.

Example 20

medium
Find the zeros of f(x)=x2+6x+5f(x) = x^2 + 6x + 5 by factoring.