If p x^2+q x+r=0 has no real roots and p, q, r are real such that p+r>0, then (A) p-q+r ≤0 (B) p

step 1 Hi, we're given here, px square plus square plus r equal to zero has no real roots. PQR are real

If p x^2+q x+r=0 has no real roots and p, q, r are real such...

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Key Concepts

Coefficient Constraints

In problems involving quadratic equations with no real roots, additional conditions on the coefficients often appear. For instance, if the quadratic has no real roots (? < 0), the sign of the leading coefficient determines whether the quadratic function is always positive or always negative. Constraints such as sums or combinations of coefficients (like a + c > 0) further refine the possible values of the coefficients, aiding in deducing inequalities or specific relations among them.

Discriminant

The discriminant of a quadratic equation is defined as ? = b² - 4ac. It is crucial because its sign indicates the nature of the roots: if ? < 0, the equation has no real roots; if ? = 0, there is exactly one real (repeated) root; and if ? > 0, there are two distinct real roots.

Quadratic Equation

A quadratic equation is a second?degree polynomial expressed in the form ax² + bx + c = 0, where a, b, and c are real coefficients and a ? 0. Its properties, such as the number and type of roots, are fundamentally determined by these coefficients, and it is a central topic in algebra.

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