Grade 9 Mathematics: Quadratic Equations

Questions and Answers

What is the value of the discriminant for the quadratic equation $x^2 + 3x - 10 = 0$?

-1

121

49 (correct)

What does it mean when the discriminant of a quadratic equation is less than zero?

The equation has exactly one real root.

The equation has two real and identical roots.

The equation has two distinct real roots.

The equation has no real roots. (correct)

If the discriminant of a quadratic equation is equal to zero, what can be concluded about the roots?

There are no real roots.

There is one real root with multiplicity two. (correct)

There is one complex root.

There are two distinct real roots.

Using the quadratic formula, what are the roots of the equation $x^2 + 3x - 10 = 0$?

2 and -5 Signup and view all the answers

Which of the following values of the coefficients would result in two distinct real roots for a quadratic equation?

a = 1, b = 5, c = 6 Signup and view all the answers

What is the standard form of a quadratic equation?

$ax^2 + bx + c = 0$ Signup and view all the answers

Which method involves rewriting the quadratic equation into a perfect square?

Completing the Square Signup and view all the answers

In the equation $x^2 - 5x + 6 = 0$, what are the solutions after factoring?

$x = 3$ and $x = 2$ Signup and view all the answers

For the equation $x^2 = 49$, what are the possible values of $x$?

$7$ and $-7$ Signup and view all the answers

What is the next step after substituting the values of $a$, $b$, and $c$ in the quadratic formula?

Simplifying the expression Signup and view all the answers

Which method is typically used to find roots when the quadratic formula is not preferred?

Completing the Square Signup and view all the answers

What does the Zero Product Property state regarding factored equations?

One of the factors must equal zero Signup and view all the answers

In the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, what does $b^2 - 4ac$ represent?

The discriminant Signup and view all the answers

Study Notes

Quadratic Equations

A quadratic equation is a polynomial of degree 2, represented in standard form as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

Methods of Solving Quadratic Equations

Extracting Square Roots
- Used for equations such as x² = 49 where taking the square root gives x = ±7.
Factoring
- Example: For the equation x² - 5x + 6 = 0, factor it to (x-2)(x-3) = 0 leading to solutions x1 = 2 and x2 = 3 using the Zero Product Property.
Completing the Square
- Involves rewriting the expression to form a perfect square. For instance, in x² + 6x - 2 = 0, it can be rewritten as (x + 3)² = 11, giving solutions x1 = -3 + √11 and x2 = -3 - √11 which approximate to 0.32 and -6.32, respectively.
Quadratic Formula
- Given by x = (-b ± √(b² - 4ac)) / 2a. Steps include rewriting the equation in standard form and substituting coefficients into the formula.
- Example with x² + 3x - 11 = 0:
  - Coefficients are a = 1, b = 3, c = -11.
  - Substituting values yields roots x1 = 2 and x2 = -5 through simplification.

Discriminant

The discriminant d = b² - 4ac determines the nature of the roots of a quadratic equation:
- d > 0: Two distinct real roots.
- d = 0: One real root with multiplicity two.
- d < 0: No real roots; roots are complex.

Description

This quiz covers the fundamental concepts of quadratic equations as part of 9th-grade mathematics. You will learn about the definition, standard form, and methods for solving quadratic equations, including extracting square roots. Test your understanding of these key topics and enhance your math skills.