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Interactive calculator Linear equations Expanding and factorising Quadratic equations Simultaneous equations GCSE and A-Level exam tips FAQs

Interactive Algebra Calculator

This page combines a practical JS calculator with a full revision guide. You can select an equation type, enter coefficients, and generate complete step-by-step working immediately. The aim is not just to produce an answer but to show every move that would gain marks in a UK classroom or exam paper. For students in Year 10, Year 11, and sixth form, this helps bridge the gap between getting the right number and writing a method that an examiner can credit. For teachers and parents, the calculator is a fast way to generate worked examples in consistent notation.

Use this tool to check your process after attempting a question yourself. If your final answer is wrong, compare each line and locate the first place your algebra changes direction. This is usually where marks are lost. In many GCSE and A-Level questions, method marks are available even if your final arithmetic is not perfect, so writing clear stages matters. The calculator is built to mirror that structure: identify known values, perform operations line by line, simplify carefully, and state the conclusion clearly.

Linear coefficients

Step-by-step working

Select an equation type, enter values, and click Show Full Working.

The calculator is intentionally method-first. Instead of returning only x = ..., it builds a sequence you can copy into exercise books and exam scripts. For linear equations it shows balancing operations. For quadratics it evaluates the discriminant, applies the quadratic formula, checks factorising where possible, and writes a completing-the-square form. For simultaneous equations it solves with elimination and then confirms through substitution. This supports students who know the topic but need more confidence in layout, sign handling, and justified transitions between lines.

Linear Equations: ax + b = c

A linear equation has one unknown and a highest power of one. In the form ax+b=c, your goal is to isolate x. The process is always the same: undo addition or subtraction first, then undo multiplication or division. Keep both sides balanced by applying each operation to both sides of the equation. This is the core principle that underpins all algebra at GCSE and continues into A-Level mechanics, statistics formulas, and pure mathematics manipulation.

Example workflow: start with 3x+5=17. Subtract 5 from both sides to get 3x=12. Divide both sides by 3 to obtain x=4. If you want to check, substitute back: 3(4)+5=12+5=17, which matches the right side, so the value is correct. This checking habit is quick and removes uncertainty before you move on. In timed exams, one substitution check can protect several marks.

When a=0, the equation stops being a standard linear equation in x. You then compare constants only. If both sides match, there are infinitely many solutions because x disappears. If constants differ, there is no solution. Recognising this edge case matters in higher-tier GCSE papers and in A-Level algebra where parameter questions appear. Examiners often test whether students notice that the structure has changed, not just whether they can perform arithmetic.

Fractions and negatives are the most common source of mistakes. If you have -2x-7=9, add 7 first to get -2x=16, then divide by -2 to get x=-8. Keep signs explicit on each line. Do not skip from the first line to the final line unless the question is very short and you are fully confident. Clear method lines are easier to audit and score.

Expanding Brackets and Factorising

Expanding and factorising are inverse skills. Expanding converts products to sums. Factorising converts sums back into products. Both appear constantly in algebraic simplification, solving quadratics, graph work, and proof style questions. The most common expansion at GCSE is FOIL for two brackets with leading coefficient 1: First, Outer, Inner, Last. For example, (x+2)(x+3)=x^2+5x+6. The middle term comes from combining the outer and inner products, which is why sign accuracy matters.

The reverse process is factorising. Given x^2-5x+6, find two numbers that multiply to +6 and add to -5. Those numbers are -2 and -3, so x^2-5x+6=(x-2)(x-3). This pattern is one of the highest-value non-calculator skills because it turns a three-term expression into roots instantly. In many exam questions, recognising factor form lets you solve faster than using the quadratic formula.

Difference of two squares: a^2-b^2=(a-b)(a+b).
Example: x^2-16=(x-4)(x+4).
This identity is useful in simplification, solving equations, and proving algebraic relationships.

Students often mis-factorise when coefficients are not 1. The safest approach is systematic: multiply the first and last coefficients, list factor pairs, and split the middle term accordingly. If factorising is not clean, switch method without delay. The best exam strategy is flexible method choice, not stubborn method commitment.

Quadratic Equations: ax^2 + bx + c = 0

Quadratic equations have degree two and can produce two roots, one repeated root, or no real roots. Three core methods are expected in UK courses: factorising, quadratic formula, and completing the square. Your calculator output includes all three perspectives where appropriate so you can compare method pathways. This is especially useful when teachers ask for one specific method but revision requires understanding of all methods.

1) Factorising method

When factors are clear, this is usually fastest. For x^2-5x+6=0, write (x-2)(x-3)=0. Then set each bracket to zero: x=2 or x=3. This direct route is common on foundation and higher GCSE, and it remains relevant in A-Level when solving intersections and motion models.

2) Quadratic formula method

The formula is x=(-b +/- sqrt(b^2-4ac))/(2a). It always works for quadratic equations and is essential when factorising is awkward or impossible over integers. The expression under the square root is the discriminant, D=b^2-4ac. If D is positive, there are two real roots. If D is zero, there is one repeated real root. If D is negative, there are no real roots (complex roots at advanced level). Writing this interpretation in words can earn communication marks in longer questions.

Keep exact forms as long as possible. For example, if D is not a perfect square, leave roots in surd form before converting to decimals only if requested. In A-Level contexts, exact surd answers are often preferred. Rounded values too early can lose accuracy and marks when substituted later into another expression.

3) Completing the square method

Completing the square rewrites the quadratic to reveal a squared binomial and a constant shift. For instance, x^2+6x+5=0 becomes (x+3)^2-4=0, so (x+3)^2=4 and then x+3=+/-2, giving x=-1 or x=-5. This method is powerful for graph interpretation because it gives the turning point immediately.

When the leading coefficient is not 1, divide through first or factor it from x-terms carefully before adding the square term. The calculator shows this structure line by line, including the value added to both sides. If your class is doing transformations of quadratic graphs, this method links algebra and graph geometry directly.

Choosing the best quadratic method in exams

Use factorising when obvious, formula when uncertain, and completing the square when the question references turning points, minimum values, or graph form. Strong students are not the ones who always use one method; they are the ones who choose efficiently and show complete logical working. In mark schemes across AQA, Edexcel, and OCR, clear method communication is repeatedly rewarded.

Simultaneous Equations: Elimination and Substitution

Simultaneous equations involve finding values of x and y that satisfy both equations at the same time. In GCSE and A-Level pure mathematics, the two standard algebraic methods are elimination and substitution. The calculator provides both approaches so you can compare them on the same question and see that they agree on one pair of values.

Elimination method

Elimination aims to remove one variable by making coefficients match. Suppose you have 2x+3y=13 and x-y=1. Multiply the second equation by 3: 3x-3y=3. Add this to the first equation and y cancels: 5x=16, so x=16/5. Substitute back to find y. In exam conditions, elimination is often quicker when coefficients are easy to align.

Substitution method

Substitution starts by isolating one variable in one equation. From x-y=1, you get x=y+1. Put this into the other equation: 2(y+1)+3y=13, giving 5y+2=13, so y=11/5, then x=16/5. This method is usually clean when a variable already has coefficient 1 or -1.

After solving, always verify in both original equations. One quick substitution check catches transcription errors and sign slips. In harder papers, simultaneous equations may produce fractions or require rearranging first. Staying calm and writing one operation per line keeps the method reliable.

GCSE AQA Edexcel OCR Exam Tips

Exam boards vary in wording, but mark schemes reward the same habits: valid algebraic steps, consistent notation, and clear final statements. For AQA, Edexcel, and OCR, do not compress five operations into one line unless they are truly straightforward. If a question says "show that", include bridging lines that justify the transformation. If a question asks for an exact value, avoid early decimal rounding. If a question asks for decimal places, round only on the final line and show your exact line just above it.

  • Write the equation before each transformation so the examiner can follow your method.
  • State the method when relevant: "Using elimination", "Using quadratic formula", or "Completing the square".
  • Keep fractions exact as long as possible.
  • Check roots by substitution when time allows.
  • For quadratics, mention the discriminant interpretation when asked about number of roots.
  • In non-calculator papers, simplify surds and fractions fully.
  • Use the answer line clearly: x = ..., y = ..., or x = ... or x = ...

For revision, focus on retrieval plus method fluency. A strong routine is: 10 linear questions, 10 expand/factorise questions, 10 quadratics mixed methods, and 5 simultaneous questions. Then mark with full working, not just final answers. This is where the calculator helps most. Enter your coefficients, compare every step, and identify whether your mistake came from algebra structure, arithmetic, or sign handling.

If you are preparing for higher-tier GCSE or A-Level, add a final stage: rewrite each solved question in the cleanest exam-ready form. This trains speed and clarity together. The fastest students are rarely those who rush blindly; they are those with a repeatable structure that avoids rework. Consistency wins marks.

Frequently Asked Questions

1) What does ax+b=c mean in GCSE algebra?

It is the standard linear equation format with one unknown, x. The letters a, b and c are constants. The key idea is balancing: whatever you do to one side must be done to the other side. First remove b by addition or subtraction, then remove a by multiplication or division. This sequence isolates x in the fewest steps. If a is zero, x may disappear, which can produce either no solution or infinitely many solutions depending on the constants.

2) When should I use the quadratic formula instead of factorising?

Factorising is best when the numbers are simple and factor pairs are obvious. The quadratic formula is safer when coefficients are awkward, when roots are irrational, or when you need guaranteed progress under time pressure. In exam practice, a sensible routine is: quick factor check, then formula if not immediate. This avoids wasting time. If the question asks for exact form, keep surds exact before any decimal approximation.

3) How do I choose elimination or substitution for simultaneous equations?

Choose elimination when coefficients can be matched quickly by multiplying equations. Choose substitution when one variable is already isolated or has coefficient 1 or -1. Both methods are valid and should give the same answer pair. If you get conflicting answers, there is an arithmetic or sign error in the lines above. The calculator outputs both styles to help you compare method logic and see where your own working diverges.

4) Why is completing the square useful if I already know the formula?

Completing the square does more than solve. It rewrites the quadratic into a form that shows the turning point and graph shift directly. This is important for interpretation questions and links algebra to graph sketching. It also improves understanding of why a quadratic has two, one, or no real roots depending on the value on the right side after squaring. Even if formula is faster for pure solving, completing the square adds conceptual depth.

5) What is the difference of two squares and where does it appear?

The identity is a^2-b^2=(a-b)(a+b). It appears in factorising questions, simplification of algebraic fractions, and proofs. A common example is x^2-25=(x-5)(x+5). Students sometimes try to use this on sums such as a^2+b^2, which does not factorise over real numbers in the same simple way. Recognition of the minus sign is critical.

6) How should I present algebra working for AQA, Edexcel and OCR?

Keep each transformation on a new line, preserve equality signs, and avoid skipping justifications. If a question says "solve", give the final value clearly. If it says "show", include enough intermediate lines to make the path unambiguous. Keep exact values where possible and round only when instructed. Use method labels when helpful. This style maximises method marks and is consistent across major UK boards even when wording differs.

7) Can this calculator help with homework without encouraging copying?

Yes, if used as a checking and reflection tool. Try the question first, write your full method, then compare against the calculator working. Focus on the first line where your method differs and correct that thinking pattern. Over time, this builds independent skill rather than answer dependence. Many students improve quickly when they track error types: sign mistakes, arithmetic slips, rearrangement errors, or missed factor pairs.