Equations and areas with the distributive law

Once you can open brackets with the extended distributive law and the short multiplication formulas, you can solve two new kinds of problems:

equations in which $\displaystyle x^{2}$ cancels once the brackets are opened;
comparing the areas of shapes whose sides are expressions.

Equations where

\displaystyle x^{2}

cancels

With a different coefficient of

\displaystyle x^{2}

on each side,

\displaystyle x^{2}

does not cancel: that is a quadratic equation, and it needs other methods.

When $\displaystyle x^{2}$ appears on both sides with the same coefficient, it cancels, and a linear equation is left. First open the brackets, then subtract $\displaystyle x^{2}$ from both sides:

\begin{aligned}(x+3)(x-2)&=x^{2}+7\\ x^{2}+x-6&=x^{2}+7 &&/-x^{2}\\ x-6&=7 &&/+6\\ x&=13\end{aligned}

Check in the original equation: $\displaystyle {(13+3)(13-2)}={16\cdot11}={176}$ and $\displaystyle {13^{2}+7}={169+7}={176}$ – equal.

Equations with the short multiplication formulas

A minus before an expression from a formula: write the expression in brackets first,

\displaystyle -(x^{2}-4x+4)

, and only then open them.

The short multiplication formulas open the brackets the same way:

\begin{aligned}(x+4)^{2}&=(x-2)^{2}+36\\ x^{2}+8x+16&=x^{2}-4x+4+36 &&/-x^{2}\\ 8x+16&=-4x+40 &&/+4x-16\\ 12x&=24 &&/:12\\ x&=2\end{aligned}

Check: $\displaystyle {(2+4)^{2}}={36}$ and $\displaystyle {(2-2)^{2}+36}={0+36}={36}$ .

Exercise 1

Open the brackets on each side, solve the equation, and check the solution by substitution.

Comparing areas

To compare two areas, subtract one from the other and simplify the difference. For example, a square’s side is $\displaystyle x$ ; one pair of its sides is lengthened by 3 and the other pair shortened by 3, which makes a rectangle.

The square’s area: $\displaystyle x^{2}$ .
The rectangle’s area: $\displaystyle {(x+3)(x-3)}={x^{2}-9}$ .
The difference: $\displaystyle {x^{2}-(x^{2}-9)}={9}$ – the square is bigger by 9, for every $\displaystyle x$ .

The growth of an area

When $\displaystyle x$ is left in the difference, the answer depends on $\displaystyle x$ . A square’s side, $\displaystyle x$ , grows by 2:

its area grows by $\displaystyle {(x+2)^{2}-x^{2}}={x^{2}+4x+4-x^{2}}={4x+4}$ ;
if the area grew by 24, then $\displaystyle {4x+4}={24}$ , so $\displaystyle {4x}={20}$ and $\displaystyle {x}={5}$ . Check: $\displaystyle {7^{2}-5^{2}}={49-25}={24}$ .

Exercise 2

Write an expression for the difference of the areas and simplify it, then answer the questions.

One rectangle is $\displaystyle x$ long and $\displaystyle x+5$ wide; another is $\displaystyle x+2$ long and $\displaystyle x+3$ wide.
Which area is bigger, and by how much?
Does the answer depend on $\displaystyle x$ ?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A square’s side is $\displaystyle x$ .
One pair of its sides is lengthened by 4 and the other pair shortened by 2, which makes a rectangle.
For which $\displaystyle x$ are the areas of the square and of the rectangle equal?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A square’s side grew by 3 cm, and its area grew by 51 cm².
What was the side at first?
Check your answer.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

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 $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$