Conditions for the type of Roots of Quadratic Equations

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Example 1
Given that the quadratic equation x2 − 6x + 7 = h(2x − 3) has two equal roots. Find the values of h.


Answer

x^2 - 6x + 7 = h(2x - 3){\rm }\,

To solve this problem, we need to arrange this quadratic equation into its general form.

x^2 - 6x + 7 = 2hx - 3h\,
x^2 - 6x - 2hx + 7 + 3h = 0\,
x^2 - (6 + 2h)x + 7 + 3h = 0\,-----(General Form)

The value of the coefficient a,b and c are:

a = 1,b = - (6 + 2h),c = 7 + 3h\,

Given that this quadratic equation has 2 equal roots, hence

b^2 - 4ac = 0\,

Substitute the values of a, b and c into the this equation:

[ - (6 + 2h)]^2 - 4(1)(7 + 3h) = 0\,
36 + 24h + 4h^2 - 28 - 12h = 0\,
4h^2 + 24h - 12h + 36 - 28 = 0\,
4h^2 + 12h + 8 = 0\,
h^2 + 3h + 2 = 0\,

Factorise the equation

(h + 1)(h + 2) = 0\,
h + 1 = 0{\rm \,or\,}h + 2 = 0\,
h = - 1{\rm \,or\,}h = - 2\,


Example 2

Example 2
Find the range of values of k if the equation x2 + 5x − (k + 3) = 0 has real roots.


Example 3
Find the range of values of p if the equation (p + 1)x2 + 2px + (p + 3) = 0 has no real root.
Example 4
Find the range of values of k such that the equation 2x2 − 8x + 2k = 0 has real roots.
Example 5
Show that the equation px2 + 8x − p + 6 = 0 has real roots for all the value of p.
Example 6
Find the range of values of p if the straight line y = 3x+p intersects the curve y = x2 + 2 at two different points.
Example 7
Find the values of m if the y = mx is the tangent to the curve x2 + y2 − 3x + 5y = 0
Example 7
Find the range of values of p if the straight line y =3x+1 does not intersects with the curve y = 2x2 + 3x − (p − 3).
Example 8
Given the equation of a curve is y = − x2 + 3x − (p + 3). Find the value of p for the following cases.

a. y is always negative
b. When y = 0, the equation has only one real root.

Example 9
If f(x) = 4x2 − kx + 9, find the range of k so that f(x) is always positive.
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