If the roots of the equation x^2-2 a x+a^2+a-3=0 are real less than 3 , then: (A) a<2 (B) 2 ≤a ≤

step 1 For what values of A, we have roots are real and less than 3. For the given equation, just work

If the roots of the equation x^2-2 a x+a^2+a-3=0 are real...

Key Concepts

Discriminant

The discriminant of a quadratic equation is the part under the square root in the quadratic formula, usually expressed as b² - 4ac. It determines whether the roots are real or complex. For the roots to be real, the discriminant must be nonnegative. This concept is essential in analyzing the nature of the roots of any quadratic equation.

Vieta's Formulas

Vieta's formulas relate the sum and product of the roots of a quadratic equation to its coefficients. Specifically, for an equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. These relationships are instrumental in deducing additional properties of the roots, such as bounding their values.

Inequalities for Quadratic Roots

When additional conditions are imposed on the values of the roots of a quadratic equation, such as stating that all roots must be less than a given number, inequality constraints are applied. This involves analyzing how the roots behave relative to the desired bound, often using the quadratic’s vertex or applying Vieta’s formulas to establish inequalities that the coefficients must satisfy.

Completing the Square

Completing the square is a method used to rewrite a quadratic expression in a form that easily reveals the vertex of the parabola represented by the equation. This technique transforms a quadratic into a perfect square plus or minus a constant, which is useful when analyzing how the roots relate to a particular value or when determining the conditions under which the quadratic satisfies certain inequalities.

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