FINDING THE X INTERCEPTS OF A QUADRATIC FUNCTION

The graph of a quadratic function is a parabola. When a parabola opens up/down, we may have a chance to have x-intercepts.

x-intercepts are the numbers on the x-axis where a parabola intersects x-axis.

Sometimes a parabola may not intersect x-axis. In that case, there are are no x-intercepts.

Consider the quadratic function given below.

y = ax2 + bx + c

Substitute f(x) = 0 in the above quadratic function to find x-intercepts.

0 = ax2 + bx + c

or

ax2 + bx + c = 0

The above is a quadratic equation and it is also called standard form of quadratic equation.

When we solve a quadratic equation, we have a chance to get two values for x. Because the degree (largest exponent) of the equation is 2.

Those two values of x are called x-intercepts.

Quadratic Function with One X-Intercept

Quadratic Function with Two X-Intercepts

Find the x intercept(s) of the quadratic functions :

Example 1 :

y = x2- 6x + 9

Solution :

y = x2  - 6x + 9

Substitute y = 0 to to find x-intercepts.

x- 6x + 9 = 0

In the quadratic equation above, the coefficient of x2 is 1. So, get two factors of the constant term '+9' such that the sum of the two factors is equal to the coefficient of x, that is '-6'.

Then the two factors of '+9' are '-3' and '-3'.

Now, split the middle term -6x using the two factors -3 and -3.

x- 6x + 9 = 0

x- 3x - 3x + 9 = 0

x(x - 3) - 3(x - 3) = 0

(x - 3)(x - 3) = 0

(x - 3)2 = 0

Taking square root on both sides,

x - 3 = 0

x = 3

The given quadratic expression has only one x-intercept and it is 3.

Example 2 :

y = x- 3x + 2

Solution :

y = x- 3x + 2

Substitute y = 0 to to find x-intercepts.

x- 3x + 2 = 0

In the quadratic equation above, the coefficient of x2 is 1. So, get two factors of the constant term '+2' such that the sum of the two factors is equal to the coefficient of x, that is '-3'.

Then the two factors of '+2' are '-1' and '-2'.

Now, split the middle term -3x using the two factors -1 and -2.

x- 3x + 2 = 0

x- 1x - 2x + 2 = 0

x(x - 1) - 2(x - 1) = 0

(x - 1)(x - 2) = 0

x - 1 = 0  or  x - 2 = 0

x = 1  or  x = 2

The given quadratic expression has two x-intercepts and they are 1 and 2.

Example 3 :

y = -9x2 + 4

Solution :

Substitute y = 0 to to find x-intercepts.

 -9x2 + 4 = 0

Multiply both sides by -1.

9x2 - 4 = 0

32x2 - 4 = 0

(3x)2 - 22 = 0

Using the identity a2 - b2 = (a + b)(a - b),

(3x + 2)(3x - 2) = 0

3x + 2 = 0  or  3x - 2 = 0

x = -2/3  or  x = 2/3

The given quadratic expression has two x-intercepts and they are -2/3 and 2/3.

Example 4 :

y = 3x- x - 2

Solution :

Substitute y = 0 to to find x-intercepts.

 3x- x - 2 = 0

In the quadratic expression above, the coefficient of x2 is 3.

Multiply the coefficient of x2 and the constant term -2.

3x(-2) = -6

Now, get two factors of '-6' such that the sum of the two factors is equal to the coefficient of x, that is '-1'.

Then the two factors of '-6' are '+2' and '-3'.

Now, split the middle term -x using the two factors +2 and -3.

 3x- x - 2 = 0

3x+ 2x - 3x - 2 = 0

x(3x + 2) - 1(3x + 2) = 0

(3x + 2)(x - 1) = 0

3x + 2 = 0  or  x - 1 = 0

x = -2/3  and  x = 1

The given quadratic function has two x-intercept and they are -2/3 and 1.

Example 5 :

y = -(x - 1)(x - 3)

Solution :

y = -(x - 1)(x - 3)

Substitute y = 0 to to find x-intercepts.

-(x - 1)(x - 3) = 0

Multiply both sides of the equation by -1.

(x - 1)(x - 3) = 0

x - 1 = 0  or  x - 3 = 0

x = 1  or  x = 3

The given quadratic function has two x-intercept and they are 1 and 3.

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