MATHIRA MATHIBAY FIRST QUARTER: Quadratic Equations Review

Flashcards cover key quadratic concepts: standard form, quadratic formula, discriminant and root nature, area problems, sum/product of roots, factoring, inequalities, and LCD considerations.

What is the standard form of a quadratic equation and what condition must the coefficient satisfy?

ax^2 + bx + c = 0 with a ≠ 0.

What is the quadratic formula?

x = (-b ± sqrt(b^2 - 4ac)) / (2a).

What is the discriminant in a quadratic equation ax^2 + bx + c = 0?

Δ = b^2 - 4ac; it determines the nature of the roots.

What does Δ > 0 imply about the roots of a quadratic equation?

Two real and distinct roots (they are rational if Δ is a perfect square).

What does Δ = 0 imply about the roots?

Two real and equal roots (a repeated root).

What does Δ < 0 imply about the roots?

Two complex (non-real) roots; no real solutions.

If a rectangle has width x and length x+4 with area 120, what is the equation for the area?

x(x+4) = 120 (or x^2 + 4x = 120).

For ax^2 + bx + c = 0, what are the sum and product of the roots?

Sum of roots = -b/a; Product of roots = c/a.

What does LCD stand for and when is it used in quadratic fraction problems?

LCD stands for Least Common Denominator; used to combine fractions with different denominators.

Solve 4x^2 = 100; what are the roots?

x = ±5.

What is a quadratic inequality?

An inequality involving a quadratic expression; solutions are intervals of x that satisfy the inequality (e.g., ax^2 + bx + c > 0).

What is the standard form of a quadratic equation in one variable?

ax^2 + bx + c = 0 (a ≠ 0).

If a = 1, b = -7, c = 12 in ax^2+bx+c=0, what is the quadratic equation?

x^2 - 7x + 12 = 0.

How do you factor the quadratic x^2 - 2x - 15 = 0?

Factor as (x - 5)(x + 3) = 0; roots x = 5 and x = -3.

What are the roots of x^2 - 6x + 9 = 0?

x = 3 (a double root).

What are the sum and product of the roots for x^2 - 6x + 9 = 0?

Sum = 6; Product = 9.

Which form demonstrates a quadratic inequality?

An inequality where the highest-degree term is squared, e.g., ax^2 + bx + c > 0.

What are the roots of x^2 + 2x - 3 = 0?

x = 1 and x = -3.

What is the sum of the roots for x^2 + 2x - 3 = 0?

Sum = -b/a = -2.

What is the area expression for a rectangle with width x and length x+4?

Area = x(x+4) = x^2 + 4x.

What are the roots of x^2 - 2x - 16 = 0?

x = 1 ± sqrt(17) (two real irrational roots).

How does the discriminant relate to rational versus irrational roots?

If Δ is a perfect square, the roots are rational; if Δ is not a perfect square (and Δ > 0), the roots are irrational; if Δ < 0, roots are complex.