Question

Discus
the
nature
of
roots
x^2+2(a+b)x+2(a^2+b^2)=0
​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

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2.6 Nature of roots (EMBFP)

Investigating the nature of roots

Use the quadratic formula to determine the roots of the quadratic equations given below and take special note of:

the expression under the square root sign and

the type of number for the final answer (rational/irrational/real/imaginary)

x2−6x+9=0

x2−4x+3=0

x2−4x−3=0

x2−4x+7=0

Choose the appropriate words from the table to describe the roots obtained for the equations above.

rational unequal real

imaginary not perfect square equal

perfect square irrational undefined

The expression under the square root, b2−4ac, is called the discriminant. Can you make a conjecture about the relationship between the discriminant and the roots of quadratic equations?

The discriminant (EMBFQ)

The discriminant is defined as Δ=b2−4ac.

This is the expression under the square root in the quadratic formula. The discriminant determines the nature of the roots of a quadratic equation. The word ‘nature’ refers to the types of numbers the roots can be — namely real, rational, irrational or imaginary. Δ is the Greek symbol for the letter D.

For a quadratic function f(x)=ax2+bx+c, the solutions to the equation f(x)=0 are given by the formula

x=−b±b2−4ac−−−−−−−√2a=−b±Δ−−√2a

If Δ<0, then roots are imaginary (non-real) and beyond the scope of this book.

If Δ≥0, the expression under the square root is non-negative and therefore roots are real. For real roots, we have the following further possibilities.

If Δ=0, the roots are equal and we can say that there is only one root.

If Δ>0, the roots are unequal and there are two further possibilities.

Δ is the square of a rational number: the roots are rational.

Δ is not the square of a rational number: the roots are irrational and can be expressed in decimal or surd form.

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Nature of roots Discriminant a>0 a<0

Roots are non-real Δ<0 d484cf42f62b770f6d60b67d88685f60.png 5ef954ace3ab3a47ba503f1b540f2823.png

Roots are real and equal Δ=0 1a3fbcbaaa98dc58c50d36ef7d1789e7.png af29fd0a2a301005ba0ed1e0a2fbd60f.png

Roots are real and unequal:

rational roots

irrational roots

Δ>0

Δ= squared rational

Δ= not squared rational

24f6df404b10a120e91bef393e7bfde2.png 592c12cf2fb349966002b4f4e78c7852.png