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Maths 9 โ€“ Algebraic expressions

๐Ÿ“˜ Notes

Algebraic Expressions: Simplify First, Then Substitute

In these questions, you must first simplify the algebraic expression, and only then substitute the given value of \(a\).

Many of these questions use expansion, factorisation, common factors, and algebraic fractions. Working carefully step by step makes the calculation much easier.

1. Main idea

  1. Simplify the expression fully.
  2. Only after simplifying, substitute the given value of \(a\).
  3. Then calculate the numerical answer.

2. Useful formulas

Square of a sum and square of a difference

\[ (a+b)^2=a^2+2ab+b^2 \] \[ (a-b)^2=a^2-2ab+b^2 \]

Difference of squares

\[ (a-b)(a+b)=a^2-b^2 \]

Common factor

\[ ax+ay=a(x+y) \]

Reducing algebraic fractions

If the numerator has a common factor with the denominator, factorise first and then cancel.

\[ \frac{a^2-9}{a+3}=\frac{(a-3)(a+3)}{a+3}=a-3 \]

3. Standard methods

Method 1: Expand brackets

Use the formulas of special products or ordinary multiplication of brackets.

Method 2: Factorise and cancel

In fractions, first factorise the numerator, then cancel common factors with the denominator.

Method 3: Collect like terms

Combine all \(a^2\)-terms, \(a\)-terms, and constants separately.

Method 4: Substitute at the end

After simplifying, replace \(a\) with the given value and calculate.

4. Worked examples

Example 1

Simplify \(a^2+6a-(a+2)^2\). Then find the value when \(a=-3\).

Solution

\[ (a+2)^2=a^2+4a+4 \] \[ a^2+6a-(a^2+4a+4) \] \[ =a^2+6a-a^2-4a-4 \] \[ =2a-4 \]

Now substitute \(a=-3\):

\[ 2(-3)-4=-6-4=-10 \]
Simplified form: \(2a-4\)
Value at \(a=-3\): \(-10\)

Example 2

Simplify \(\dfrac{2a^2-6a}{a-3}-4a+8\). Then find the value when \(a=4\).

Solution

\[ 2a^2-6a=2a(a-3) \] \[ \frac{2a(a-3)}{a-3}-4a+8 \] \[ =2a-4a+8 \] \[ =-2a+8 \]

Now substitute \(a=4\):

\[ -2(4)+8=-8+8=0 \]
Simplified form: \(-2a+8\)
Value at \(a=4\): \(0\)

Example 3

Simplify \((a-3)^2-a(a-3)\). Then find the value when \(a=-4\).

Solution

\[ (a-3)^2=a^2-6a+9 \] \[ a(a-3)=a^2-3a \] \[ (a^2-6a+9)-(a^2-3a) \] \[ =a^2-6a+9-a^2+3a \] \[ =-3a+9 \]

Now substitute \(a=-4\):

\[ -3(-4)+9=12+9=21 \]
Simplified form: \(-3a+9\)
Value at \(a=-4\): \(21\)

Example 4

Simplify \(\dfrac{2a^3-4a^2}{a^2}-7a+3\). Then find the value when \(a=3\).

Solution

\[ \frac{2a^3-4a^2}{a^2}=\frac{a^2(2a-4)}{a^2}=2a-4 \] \[ 2a-4-7a+3 \] \[ =-5a-1 \]

Now substitute \(a=3\):

\[ -5(3)-1=-15-1=-16 \]
Simplified form: \(-5a-1\)
Value at \(a=3\): \(-16\)

Example 5

Simplify \((a-5)(a+5)-a(a-4)\). Then find the value when \(a=-5\).

Solution

\[ (a-5)(a+5)=a^2-25 \] \[ a(a-4)=a^2-4a \] \[ (a^2-25)-(a^2-4a) \] \[ =a^2-25-a^2+4a \] \[ =4a-25 \]

Now substitute \(a=-5\):

\[ 4(-5)-25=-20-25=-45 \]
Simplified form: \(4a-25\)
Value at \(a=-5\): \(-45\)

Example 6

Simplify \(\dfrac{a^2}{a+1}\cdot \dfrac{3a+3}{a}-a-8\). Then find the value when \(a=3\).

Solution

\[ 3a+3=3(a+1) \] \[ \frac{a^2}{a+1}\cdot \frac{3(a+1)}{a} \] \[ =3a \] \[ 3a-a-8 \] \[ =2a-8 \]

Now substitute \(a=3\):

\[ 2(3)-8=6-8=-2 \]
Simplified form: \(2a-8\)
Value at \(a=3\): \(-2\)

5. Common mistakes

  • Substituting the value of \(a\) too early, before simplifying.
  • Expanding \((a+b)^2\) or \((a-b)^2\) incorrectly.
  • Forgetting brackets when subtracting an expression.
  • Cancelling terms instead of cancelling common factors.
  • Making sign mistakes when collecting like terms.

6. Summary

\[ (a+b)^2=a^2+2ab+b^2 \] \[ (a-b)^2=a^2-2ab+b^2 \] \[ (a-b)(a+b)=a^2-b^2 \]

In these questions, simplify first and substitute second. This makes the work shorter, cleaner, and much easier to calculate.