A partial product refers to any of the multiplication results that lead up to an overall multiplication result.

A binomial is an algebraic expression representing the sum or difference of exactly two unlike terms.

To factorise is to express a term as the product of its factors.

A solution to an equality with one variable is a value for the variable which, when substituted, maintains the equality between the expressions.

Problem solving with algebraic manipulation

Maths

In this unit, pupils learnt to solve quadratic equations by factorising and this skill will be required in the future unit through manipulating two simultaneous equations to find the solutions. For pupils studying the higher content, solving quadratic equations is required if one of the simultaneous equations is a quadratic.

In the prior unit, pupils used the distributive law to find the product of two binomials. In algebraic manipulation, pupils use apply this knowledge in order to factorise quadratics.

Algebraic manipulation

A right triangle has area $$24 cm^2$$ and side lengths $$x$$ and $$x + 2$$. What is the value of $$x$$?

The sum of two consecutive odd integers is 156. What are the two integers?

What is the positive value of $$x$$ where $$(x + 4)(x - 2) = 0$$?

The difference between the square of a positive integer and the integer itself is 110. What is the integer?

The sum of the squares of two consecutive positive integers is 365. What is the smallest integer?

The product of two consecutive integers is 272. What are the two integers?

Evaluate the expression $$\frac{3x}{2x + 5}$$ for $$x = 5$$. 

Evaluate the expression $$\frac{x}{x + 1}$$ for $$x = 19$$. Give your answer as a decimal.

A book store sells a book for £25, which is 25% more than its cost price. Which of these calculates cost price, $$p$$, of the book?

Expand and simplify $$4(x - 2) - 2(3x + 4)$$

Simplify the expression $$\frac{x^2 - 9}{x - 3}$$

Fully factorise the expression $$2x^2 + 12x$$

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