Question

if 1+√2
is a root of a quadratic equation will rational coefficients, write its other root.
4x +3=0. and hence find the nature​

Answer 1

Answer:

= (-b-√b2-4ac)/2a and β = (-b+√b2-4ac)/2a

Here a, b, and c are real and rational. Hence, the nature of the roots α and β of equation ax2 + bx + c = 0 depends on the quantity or expression (b2 – 4ac) under the square root sign. We say this because the root of a negative number can’t be any real number. Say x2 = -1 is a quadratic equation. There is no real number whose square is negative. Therefore for this equation, there are no real number solutions.

Nature of Roots

Hence, the expression (b2 – 4ac) is called the discriminant of the quadratic equation ax2 + bx + c = 0. Its value determines the nature of roots as we shall see. Depending on the values of the discriminant, we shall see some cases about the nature of roots of different quadratic equations.

quadratic equation cheat sheet

Nature Of Roots

Let us recall the general solution, α = (-b-√b2-4ac)/2a and β = (-b+√b2-4ac)/2a

Case I: b2 – 4ac > 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive, then the roots α and β of the quadratic equation ax2 +bx+ c = 0 are real and unequal.

Case II: b2– 4ac = 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is zero, then the roots α and β of the quadratic equation ax2+ bx + c = 0 are real and equal.

Case III: b2– 4ac < 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is negative, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.

Case IV: b2 – 4ac > 0 and perfect square

When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational and unequal.

Case V: b2 – 4ac > 0 and not perfect square

When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal.

Here the roots α and β form a pair of irrational conjugates.

Case VI: b2 – 4ac > 0 is perfect square and a or b is irrational

When a, b, and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax2 + bx + c = 0 are irrational.

Let us just summarize all the above cases in this table below:

b2 – 4ac > 0 Real and unequal

b2 – 4ac = 0 Real and equal

b2 – 4ac < 0 Unequal and Imaginary

b2 – 4ac > 0 (is a perfect square) Real, rational and unequal

b2 – 4ac > 0 (is not a perfect square) Real, irrational and unequal

b2 – 4ac > 0 (is aperfect square and a or b is irrational) Irrational