Grade 10 Mathematics: Nature of Roots and Equations Reducible to Quadratic Form

A set of question-and-answer flashcards covering Grade 10 Mathematics concepts on the discriminant, nature of quadratic roots, and solving equations reducible to quadratic form.

What is the standard form of a quadratic equation and its key condition for the coefficient $a$?

The standard form is $ax^2 + bx + c = 0$, where $a \neq 0$.

What is the quadratic formula used to solve quadratic equations?

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

What is the discriminant of a quadratic equation and what algebraic expression defines it?

The discriminant is the expression $b^2 - 4ac$ located inside the radical of the quadratic formula.

What is the main purpose of calculating the discriminant of a quadratic equation?

It determines the nature of the roots without having to solve the quadratic equation completely.

What is the nature of the roots when the discriminant is greater than zero ($D > 0$) and a perfect square?

The quadratic equation has two real, unequal, and rational roots.

What is the nature of the roots when the discriminant is greater than zero ($D > 0$) and not a perfect square?

The quadratic equation has two real, unequal, and irrational roots.

What is the nature of the roots when the discriminant is equal to zero ($D = 0$)?

The quadratic equation has two real and equal roots.

What is the nature of the roots when the discriminant is less than zero ($D < 0$)?

The quadratic equation has no real roots.

How is the nature of the roots determined for the quadratic equation $2x^2 - 4x - 1 = 0$?

The discriminant is $24$ from $(-4)^2 - 4(2)(-1) = 16 + 8 = 24$. Since $24 > 0$ and is not a perfect square, the roots are real, unequal, and irrational.

How is the nature of the roots determined for the quadratic equation $x^2 - 6x + 9 = 0$?

The discriminant is $0$ from $(-6)^2 - 4(1)(9) = 36 - 36 = 0$, which indicates two real and equal roots.

How is the nature of the roots determined for the quadratic equation $x^2 + 4x + 8 = 0$?

The discriminant is $-16$ from $4^2 - 4(1)(8) = 16 - 32 = -16$. Since $-16 < 0$, there are no real roots.

What general algebraic form allows an equation to be reduced to quadratic form using substitution?

An equation in the form $ax^{2n} + bx^n + c = 0$, which reduces to $au^2 + bu + c = 0$ by letting $u = x^n$.

What are the solutions for $x$ in the reducible equation $x^4 - 13x^2 + 36 = 0$?

By letting $u = x^2$, the equation becomes $u^2 - 13u + 36 = 0$, factoring into $(u - 4)(u - 9) = 0$ to yield $u = 4$ and $u = 9$. Solving $x^2 = 4$ and $x^2 = 9$ gives $x = \pm 2$ and $x = \pm 3$.

What are the real solutions for $x$ in the reducible equation $x^6 - 9x^3 + 8 = 0$?

By letting $u = x^3$, the equation becomes $u^2 - 9u + 8 = 0$, factoring into $(u - 1)(u - 8) = 0$ to yield $u = 1$ and $u = 8$. Taking the cube roots gives $x = 1$ and $x = 2$.