We already acquainted with the general form of quadratic expression ax^2 + bx + c now we will discuss about the sign of the quadratic expression ax^2 + bx + c = 0 (a ≠ 0).
When x be real then, the sign of the quadratic expression ax^2 + bx + c is the same as a, except when the roots of the quadratic equation ax^2 + bx + c = 0 (a ≠ 0) are real and unequal and x lies between them.
Proof:
We know the general form of quadratic equation ax^2 + bx + c = 0 (a ≠ 0) ..................... (i)
Let α and β be the roots of the equation ax^2 + bx + c = 0 (a ≠ 0). Then, we get
α + β = -b/a and αβ = c/a
Now, ax^2 + bx + c = a(x^2 + b/a x + c/a)
= a[x^2 - (α + β)x + αβ]
= a[x(x - α) - β(x - α)]
or, ax^2 + bx + c = a(x - α)(x - β) ..................... (ii)
Case I:
Let us assume that the roots α and β of equation ax^2 + bx + c = 0 (a ≠ 0) are real and unequal and α > β. If x be real and β < x < α then,
x - α < 0 and x - β > 0
Therefore, (x - α)(x - β) < 0
Therefore, from ax^2 + bx + c = a(x - α)(x - β) we get,
ax^2 + bx + c > 0 when a < 0
and ax^2 + bx + c < 0 when a > 0
Therefore, the quadratic expression ax^2 + bx + c has a sign of opposite to that of a when the roots of ax^2 + bx + c = 0 (a ≠ 0) are real and unequal and x lie between them.
Case II:
Let the roots of the equation ax^2 + bx + c = 0 (a ≠ 0) be real and equal i.e., α = β.
Then, from ax^2 + bx + c = a(x - α)(x - β) we have,
ax^2 + bx + c = a(x - α)^2 ................ (iii)
Now, for real values of x we have, (x - α)^2 > 0.
Therefore, from ax^2 + bx + c = a(x - α)^2 we clearly see that the quadratic expression ax^2 + bx + c has the same sign as a.
Case III:
Let us assume α and β be real and unequal and α > β. If x is real and x < β then,
x - α < 0 (Since, x < β and β < α) and x - β < 0
(x - α)(x - β) > 0
Now, if x > α then x – α >0 and x – β > 0 ( Since, β < α)
(x - α)(x - β) > 0
Therefore, if x < β or x > α then from ax^2 + bx + c = a(x - α)(x - β) we get,
ax^2 + bx + c > 0 when a > 0
and ax^2 + bx + c < 0 when a < 0
Therefore, the quadratic expression ax^2 + bx + c has the same sign as a when the roots of the equation ax^2 + bx + c = 0 (a ≠ 0) are real and unequal and x does not lie between them.
Case IV:
Let us assume the roots of the equation ax^2 + bx + c = 0 (a ≠ 0) be imaginary. Then we can take, α = p + iq and β = p - iq where p and q are real and i = √-1.
Again from ax^2 + bx + c = a(x - α)(x - β) we get
ax^2 + bx + c = a(x - p - iq)(x - p + iq)
or, ax^2 + bx + c = a[(x – p)^2 + q^2] .....................(iv)
Hence, (x - p)^2 + q^2 > 0 for all real values of x (Since, p, q are real)
Therefore, from ax^2 + bx + c = a[(x - p)^2 + q^2] we have,
ax^2 + bx + c > 0 when a > 0
and ax^2 + bx + c < 0 when a < 0.
Therefore, for all real values of x from the quadratic expression ax^2 + bx + c we get the same sign as a when the roots of ax^2 + bx + c = 0 (a ≠ 0) are imaginary.
Notes:
(i) When the discriminant b^2 - 4ac = 0 then the roots of the quadratic equation ax^2 + bx + c = 0 are equal. Therefore, for all real x, the quadratic expression ax^2 + bx + c becomes a perfect square when discriminant b^2 -4ac = 0.
(ii) When a, b are c are rational and discriminant b^2 - 4ac is a positive perfect square the quadratic expression ax^2 + bx + c can be expressed as the product of two linear factors with rational coefficients.
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