Practice Maths

Mixed Algebraic Problems

Key Ideas

Skills covered in this lesson: expanding brackets, factorising (common factor and difference of squares), solving linear equations, solving quadratic equations, substitution, rearranging formulas

• Expanding: Distribute each term — e.g. 3(2x − 5) = 6x − 15; (x + 3)(x − 4) = x² − x − 12.
• Factorising (HCF): e.g. 6x² − 9x = 3x(2x − 3).
• Factorising (DOPS): a² − b² = (a + b)(a − b) — e.g. x² − 25 = (x + 5)(x − 5).
• Solving linear equations: isolate the variable using inverse operations.
• Solving quadratics: factorise then apply the null factor law, or use the quadratic formula.
• Substitution and rearranging: replace variables with values, or change the subject before evaluating.
Strategy Before working, identify which technique each question needs. Key signals: “expand”/“simplify” → expanding; “factorise” → factorising; “solve” → equation solving; “make _ the subject” → rearranging; “find when” with a formula → substitution.

Overview of Year 9 Algebra Skills

By this point you have built a comprehensive algebra toolkit. This lesson brings it all together. The key skills are: expanding brackets (including binomial products), factorising (common factor and difference of two squares), solving equations (linear and some quadratic), substitution into expressions and formulas, and rearranging formulas. Mixed problems draw on several of these at once.

Expanding and Factorising

Expanding: Use the distributive law. For binomials, use FOIL: (a + b)(c + d) = ac + ad + bc + bd.

Example: (2x + 3)(x − 4) = 2x2 − 8x + 3x − 12 = 2x2 − 5x − 12.

Factorising: First check for a highest common factor (HCF). Then look for special patterns: difference of two squares a2 − b2 = (a + b)(a − b), or factorise a trinomial by finding two numbers that multiply to give c and add to give b in ax2 + bx + c.

Example: Factorise x2 − 7x + 12. Find two numbers that multiply to 12 and add to −7: they are −3 and −4. So x2 − 7x + 12 = (x − 3)(x − 4).

Solving Equations

For linear equations: collect like terms, then use inverse operations to isolate x. Check your solution by substituting back.

For equations with fractions: multiply every term by the LCM of denominators to clear fractions first.

For quadratic equations (when factorising is required): factorise, set each factor equal to zero, solve. Example: x2 − 7x + 12 = 0 ⇒ (x − 3)(x − 4) = 0 ⇒ x = 3 or x = 4.

Multi-Step Algebraic Problems

Multi-step problems might ask you to: expand an expression, then substitute a value; or rearrange a formula, then use it to solve for a variable; or set up an equation from a worded situation, solve it, then interpret the answer.

Strategy: (1) Read carefully and identify what is being asked. (2) Decide which algebraic technique is required. (3) Work in logical steps, showing all working. (4) Check that your final answer makes sense in context.

Example: A rectangle has length (2x + 3) and width (x − 1). Its area is 40 cm2. Find x. Set up: (2x + 3)(x − 1) = 40. Expand: 2x2 + x − 3 = 40. Rearrange: 2x2 + x − 43 = 0. Solve by quadratic formula or trial.

Exam-Style Consolidation Tips

When tackling mixed algebra in an exam: (1) Show all working — method marks are available even if the final answer is wrong. (2) Present your solution in a logical order. (3) Label steps clearly. (4) Substitute back to check. (5) Don't rush expanding brackets — sign errors here cause cascading mistakes through the whole solution.

Key tip: The most common errors in mixed algebra are sign errors when expanding brackets and forgetting to set the quadratic equal to zero before factorising to solve it. Check: after factorising, did you write "= 0" and find two solutions? If the equation isn't set to zero first, the zero-product property doesn't apply.

Mastery Practice

  1. Identify and apply the correct technique for each part. State the technique you are using. Fluency

    1. Expand and simplify: 4(3x − 2) + 5x
    2. Expand and simplify: (x + 6)(x − 2)
    3. Factorise: 12x² − 8x
    4. Factorise: x² − 49
    5. Solve: 5x + 3 = 28
    6. Solve: 3(2x − 1) = 21
    7. Factorise and solve: x² − 5x + 6 = 0
    8. Evaluate: 2x² − 3x + 1 when x = −2

  2. Continue identifying and applying the correct technique. Fluency

    1. Expand: (2x + 3)(3x − 5)
    2. Factorise: x² − 3x − 10
    3. Solve: 2x + 1⁄3 = 5
    4. Make y the subject: 3x + 2y = 12
    5. Expand and simplify: (x + 4)² − (x − 3)²
    6. Factorise: 4x² − 36
    7. Solve: x² + x − 12 = 0
    8. Evaluate: A = πr² when r = 4 (leave in terms of π, then round to 1 d.p.)

  3. These questions require you to connect two or more algebra techniques. Understanding

    1. The area of a rectangle is x² + 7x + 12. Factorise this expression to find possible expressions for the length and width.
    2. Expand (x + a)(x + b) and show that for any values of a and b, the result can be written as x² + (a + b)x + ab. Use this to quickly factorise x² + 9x + 20.
    3. Rearrange y = 3x + 2 to make x the subject. Hence find x when y = 11.
    4. Show that (x + 5)² − (x + 2)² simplifies to a linear expression, then solve it equal to 21.

  4. Justify your reasoning. Each question has an error or a deeper thinking aspect. Understanding

    1. A student says (x + 3)² = x² + 9. Explain the mistake and give the correct expansion.
    2. A student factorises x² − 4x as x(x − 4) and says the only solution to x² − 4x = 0 is x = 4. What error did they make?
    3. For the formula A = ½bh, a student rearranges to get h = A − ½b. Find the error and show the correct rearrangement.
    4. Show that (2x + 3)(2x − 3) is a difference of two perfect squares and evaluate it when x = 5.

  5. Multi-step problems requiring several algebra skills. Show all working. Problem Solving

    1. A rectangular garden has a length that is 3 more than twice its width. The perimeter is 42 m.
      1. Write an expression for the length in terms of w.
      2. Form an equation using P = 2(l + w) = 42 and solve for w.
      3. Hence find the length and the area of the garden.
    2. The product of two consecutive odd integers is 143. Let the smaller integer be n.
      1. Write an equation in terms of n.
      2. Expand and rearrange to form a quadratic equation.
      3. Solve to find the two integers.
    3. A ball is thrown upward and its height (m) after t seconds is h = −5t² + 15t.
      1. Factorise the expression for h.
      2. When does the ball hit the ground? (Solve h = 0.)
      3. Find the height at t = 1.5 s by substitution. What does this suggest about the symmetry of the path?

  6. Each part requires a different algebra technique. Identify which technique applies, then solve. Show all working. Problem Solving

    1. Expand and simplify: (3x − 4)² + 2(x + 1).
    2. Factorise completely: 2x³ − 8x.
    3. Solve: 3x⁄(x − 2) = 6. (Hint: multiply both sides by (x − 2) first.)
    4. Make r the subject of V = &frac43;πr³, then find r to 1 d.p. when V = 113.1 cm³ (use π = 3.14).

  7. Each piece of working below contains exactly one error. Identify the error, explain what went wrong, and give the correct answer. Problem Solving

    1. Expand (x − 5)²: Student writes x² − 25.
      What is the correct expansion?
    2. Factorise x² + 5x + 6: Student writes (x + 2)(x + 4).
      What is the correct factorisation?
    3. Solve 2x² = 50: Student divides by 2 to get x² = 25, then writes x = 5 only.
      What solution is missing and why?
    4. Rearrange s = ut + ½at² to make u the subject: Student writes u = s − ½at².
      What is the correct rearrangement?

  8. These problems require you to choose and chain algebra techniques. Plan your approach before starting. Problem Solving

    1. The area of a square is (x² + 10x + 25) cm².
      1. Factorise to find the side length in terms of x.
      2. If x = 3, find the side length and the perimeter.
    2. A rectangular paddock has an area of x² + x − 6 m² and a width of (x − 2) m.
      1. Factorise to find the length.
      2. If x = 5, find the area numerically and verify it matches the dimensions.
    3. Show that (x + 3)² − 9 can be factorised as x(x + 6). Hence solve x(x + 6) = 0.

  9. Real-world multi-step algebra. Show full reasoning. Problem Solving

    1. A frame surrounds a picture. The picture is x cm wide and (x + 4) cm tall. The frame adds a 3 cm border on all sides.
      1. Write an expression for the total area of the picture plus frame.
      2. Expand and simplify.
      3. If x = 10, find the total area.
    2. The sum of a number and its square is 42. Let the number be n.
      1. Write an equation.
      2. Rearrange to standard form and factorise.
      3. Find all possible values of n.
    3. A ball is dropped from a building. Its height (m) after t seconds is h = 80 − 5t².
      1. When does the ball hit the ground?
      2. Make t the subject of the formula.
      3. At what time is the ball at 35 m?

  10. For each question, name the algebra technique you will use, then solve. Justify your choice of technique in one sentence. Problem Solving

    1. (2x + 5)² − (2x − 5)². Simplify and evaluate when x = 3.
    2. Factorise 3x² − 27 and hence solve 3x² − 27 = 0.
    3. Solve (x + 2)⁄4 = (2x − 1)⁄3. (Hint: cross-multiply.)
    4. A formula for the area of an annulus (ring) is A = π(R² − r²), where R is the outer radius and r the inner radius.
      1. Factorise the expression R² − r².
      2. Make R the subject of the formula.
      3. Find R (to 1 d.p.) when A = 75.4 cm² and r = 3 cm (use π = 3.14).
See Answers ➔