Algebra · step 4 of 13

Solving linear & quadratic equations

A linear equation has exactly one solution unless it degenerates; a quadratic has two, one or none over ℝ depending on the discriminant b² − 4ac. Three methods solve quadratics — factoring when the roots are rational, completing the square when you need the vertex, and the formula always — and knowing which to reach for saves most of the time.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Rearrange to standard formGet ax² + bx + c = 0 with everything on one side. Solving before this is where sign errors creep in.
Try factoring firstIf ac splits into two numbers summing to b, factoring is fastest and exact.
Compute the discriminantΔ = b² − 4ac. Positive gives two real roots, zero gives one repeated root, negative gives none over ℝ.
Apply the formula or complete the squarex = (−b ± √Δ)/(2a). Completing the square additionally reveals the vertex, which factoring does not.

Worked example

Solve 2x² − 7x + 3 = 0.

ac = 2·3 = 6; two numbers multiplying to 6 and adding to −7 are −1 and −6.
Split: 2x² − 6x − x + 3 = 0.
Group: 2x(x − 3) − 1(x − 3) = 0, giving (2x − 1)(x − 3) = 0.
Each factor zero: x = 1/2 or x = 3. Check with the discriminant: 49 − 24 = 25, √25 = 5 ✓.

Answer. x = 1/2 or x = 3.

Where marks get dropped

These are the specific errors that cost credit on solving linear & quadratic equations questions — QED's rubric penalises each of them separately.

Dividing both sides by x, which loses the root x = 0. Factor instead: x² = 3x becomes x(x − 3) = 0 with roots 0 and 3.
Applying the zero-product rule to a non-zero right-hand side. (x−1)(x−2) = 6 does not give x = 7 or x = 8.
Mis-signing −b in the formula. When b is negative, −b is positive, and this is the most common arithmetic slip.

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Solving linear & quadratic equations — frequently asked questions

When should I complete the square?

When you need the vertex, the maximum or minimum, or a standard form for a circle or conic. It is also how the quadratic formula is derived.

What does a negative discriminant mean?

No real roots — the parabola misses the x-axis. Over ℂ there are still two roots, a complex conjugate pair.

How do I check my answers?

Substitute back into the ORIGINAL equation, not a rearranged version. Alternatively check that the roots sum to −b/a and multiply to c/a.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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