Question

find zeros p(x)=X²-5x+4​

Answer 1

Given:

p(x) = x² - 5x + 4

To find:

The zeroes of p(x).

Solution:

First,

To check that p(x) = x² - 5x + 4 has real roots,

a = 1, b = -5, c = 4

b² - 4ac

(-5)² - 4 * 1 * 4

25 - 16

9

Since, 9 > 0, p(x) has 2 real roots.

Now,

We know that:

-b ± √ b² - 4ac / 2a tto find zeroes of a given polynomial p(x)

-(-5) ± √ (-5)² - 4 * 1 * 4 / 2 * 1

5 ± √ 25 - 16 / 2

5 ± √9 / 2

5 ± 3 / 2

5 + 3 / 2 and 5 - 3 / 2

8 / 2 and 2 / 2

4 and 1 are the zeroes of p(x).

Conclusion:

Therefore, the roots of p(x) = x² - 5x + 4 are 4 and 1.

Verification:

p(x) = x² - 5x + 4

p(1) = 1² - 5 * 1 + 4

p(1) = 1 - 5 + 4

p(1) = 5 - 5

p(1) = 0

p(4) = 4² - 5 * 4 + 4

p(4) = 16 - 20 + 4

p(4) = 20 - 20

p(4) = 0

Since, p(1) = 0 and p(4) = 0.

Therefore, 1 and 4 are the zeroes of p(x) = x² - 5x + 4.

Extra Information:

To find the zeroes of a quadratic polynomial we use,

-b ± √ b² - 4ac / 2a

To check whether a quadratic polynomial has zeroes we can check by the following ones,

b² - 4ac > 0, it has 2 real zeroes

b² - 4ac = 0, it has equal zeroes

b² - 4ac < 0, it has no real zeroes

Answer 2

Given:

p(x) = x² - 5x + 4

To find:

The zeroes of p(x).

Solution:

First,

To check that p(x) = x² - 5x + 4 has real roots,

a = 1, b = -5, c = 4

b² - 4ac

(-5)² - 4 * 1 * 4

25 - 16

9

Since, 9 > 0, p(x) has 2 real roots.

Now,

We know that:

-b ± √ b² - 4ac / 2a tto find zeroes of a given polynomial p(x)

-(-5) ± √ (-5)² - 4 * 1 * 4 / 2 * 1

5 ± √ 25 - 16 / 2

5 ± √9 / 2

5 ± 3 / 2

5 + 3 / 2 and 5 - 3 / 2

8 / 2 and 2 / 2

4 and 1 are the zeroes of p(x).

Conclusion:

Therefore, the roots of p(x) = x² - 5x + 4 are 4 and 1.

Verification:

p(x) = x² - 5x + 4

p(1) = 1² - 5 * 1 + 4

p(1) = 1 - 5 + 4

p(1) = 5 - 5

p(1) = 0

p(4) = 4² - 5 * 4 + 4

p(4) = 16 - 20 + 4

p(4) = 20 - 20

p(4) = 0

Since, p(1) = 0 and p(4) = 0.

Therefore, 1 and 4 are the zeroes of p(x) = x² - 5x + 4.

Know more :

To find the zeroes of a quadratic polynomial we use,

-b ± √ b² - 4ac / 2a

To check whether a quadratic polynomial has zeroes we can check by the following ones,

b² - 4ac > 0, it has 2 real zeroes

b² - 4ac = 0, it has equal zeroes

b² - 4ac < 0, it has no real zeroes