Review: the whole course

Algebra, grade 9 Lesson 26 of 26Practice

Review: the whole course

Simplify with the laws of powers.

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Simplify.

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Open the brackets and simplify.

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Calculate without a calculator, with the short multiplication formulas.
Write $\displaystyle {31}={30+1}$ , $\displaystyle {29}={30-1}$ first.

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Solve the equation $\displaystyle {(x+4)(x-1)}={x^{2}+5}$ .
Open the brackets on the left side first.

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The side of a square is $\displaystyle a$ cm ( $\displaystyle a>3$ ).
A rectangle’s sides are $\displaystyle a+3$ cm and $\displaystyle a-3$ cm.
Which has the greater area, and by how much?
Calculate the difference of the areas.

The difference

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Factor.

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Solve by factoring.

Write the domain of each expression, then simplify it.

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Solve.
Write the domain, and check each solution against it.

Here is the graph of a function $\displaystyle f(x)$ .
Find from the graph:
a. the intersection points with the axes
b. the positive and the negative domains
c. the extremum points, and whether each is a minimum or a maximum, local or absolute
d. the increasing and the decreasing domains
e. how many solutions the equation $\displaystyle {f(x)}={-2}$ has
f. the extremum points of the function $\displaystyle f(x)+4$

\displaystyle y=x^{3}-3x-2

\displaystyle {f(x)}={-2}

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The parabola is $\displaystyle {y}={-2(x-1)^{2}+8}$ .
Find the vertex, and whether it is a minimum or a maximum; the axis of symmetry; the intersection points with the axes; the increasing and the decreasing domains; the positive and the negative domains.
Sketch the parabola.

a. For $\displaystyle {y}={2(x+1)(x-3)}$ : find the intersection points with the $\displaystyle x$ -axis and the vertex, then write it in the standard form.
b. For $\displaystyle {y}={x^{2}-6x+5}$ : find the intersection point with the $\displaystyle y$ -axis and the vertex, then write it in the product form.

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Solve with the quadratic formula.

Solve the inequalities.

Solve the system $\displaystyle {y}={x^{2}-2x}$ , $\displaystyle {y}={x+4}$ algebraically.
Check the solutions on the graph (the line is dashed).

\displaystyle y=x^{2}-2x

\displaystyle y=x+4

A rectangle is 4 cm longer than it is wide, and its area is 60 cm².
Find its sides.

The width

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The length

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The sum of the squares of two consecutive natural numbers is 85.
What are the numbers?

The smaller number

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The greater number

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Formulas: algebra, grade 9

Open as a page

The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$