Grade 9 Mathematics First Quarter Exam

Questions and Answers

Which of the following quadratic equations has a root of 6 and -3?

x^2 - 3x - 18 = 0

x^2 + 6x - 18 = 0 (correct)

x^2 + 3x - 18 = 0

x^2 - 6x + 18 = 0

What are the roots of the equation x^2 - 5x - 36 = 0?

-6 and 6

9 and -4 (correct)

12 and -3

6 and -6

Which of the following inequalities has the solution x < -3 or x > 4?

x^2 - 5x + 6 > 0

x^2 + 6x + 9 > 0

x^2 - 2x - 3 > 0 (correct)

x^2 - 2x - 24 > 0

Which quadratic equation has a root of -6 and 3?

x^2 + 6x - 18 = 0 Signup and view all the answers

Which equation can be transformed into a quadratic equation?

3x(x + 4) = 10 Signup and view all the answers

Which method is most efficient for quickly solving the equation $49x^2 = 64$?

Extracting Square Roots Signup and view all the answers

What is the standard form of a polynomial equation of degree two?

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

Which of the following best characterizes the term 'roots' in reference to quadratic equations?

Solutions Signup and view all the answers

If one of the roots of the equation $x^2 - 7x + 12 = 0$ is 3, what is the other root?

4 Signup and view all the answers

What is the relationship between the roots and the coefficients in a quadratic equation?

The sum of the roots equals $-b/a$ and the product equals $c/a$. Signup and view all the answers

Which values of $x$ satisfy the equation $x^2 + 11x - 26 = 0$?

13 and 2 Signup and view all the answers

Which statement accurately describes a quadratic equation?

It may have two equal roots, one root, or no real roots. Signup and view all the answers

In the polynomial equation $ax^2 + bx + c = 0$, what is the condition for $a$?

$a$ cannot be equal to $0$. Signup and view all the answers

Study Notes

Quadratic Equations Overview

A polynomial equation of degree two is expressed as ( ax^2 + bx + c = 0 ) where ( a \neq 0 ).
Characteristics of quadratic equations include solutions known as roots, which can be real or complex.

Solving Quadratic Equations

Extracting square roots is an efficient method to solve equations like ( 49x^2 = 64 ).
Completing the square is another method but may require more steps compared to extracting square roots.

Key Concepts

Solutions of a quadratic equation are referred to as roots or zeros.
The relationship between the roots and coefficients can be summarized:
- The sum of the roots equals ( -\frac{b}{a} ).
- The product of the roots equals ( \frac{c}{a} ).

Finding Roots

To find the other root of ( x^2 - 7x + 12 = 0 ) given one root is 3, use the factorization method or apply the quadratic formula. The other root is 4.
If a quadratic equation is structured as ( (x - r_1)(x - r_2) = 0 ), roots ( r_1 ) and ( r_2 ) can be directly used to form the equation.

Identifying Roots from Equations

Equations can yield values that satisfy them. For example, testing potential roots like -13, 13, and 2 in equations ( x^2 + 11x - 26 = 0 ) and ( x^2 + 15x + 26 = 0 ) helps identify valid solutions.
Roots like -6 and 3 can derive the equation ( (x + 6)(x - 3) = 0 ).

Inequalities and Solution Sets

Inequalities involving quadratic expressions, such as ( x^2 - 2x - 3 > 0 ), require finding critical points and testing intervals.
The solution set of ( x^2 - 5x ≤ 0 ) may include combinations of values or intervals satisfying the inequality.

Transformations and Quadratics

Certain contextual equations, such as ( 3x(x + 4) = 10 ), can be transformed into a quadratic form for easier solving.
Proper inequality handling is crucial for determining valid solution sets and ranges.

Properties of Quadratic Functions

The graph of a quadratic function, represented as a parabola, opens upwards if ( a > 0 ) and downwards if ( a < 0 ).
The vertex, where the parabola turns, can provide minimum or maximum values.

Summary of Key Quadratic Equation Types

Standard Form: ( ax^2 + bx + c = 0 )
Vertex Form: ( a(x-h)^2 + k ), where ( (h, k) ) is the vertex.
Factored Form: ( a(x - r_1)(x - r_2) = 0 ), where ( r_1 ) and ( r_2 ) are the roots.

These concepts provide foundational knowledge for grade 9 mathematics in understanding and solving quadratic equations and their properties.