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neurologyYou need to know

• A two-step linear equation has the form `ax + b = c` with `a ≠ 0`, and its solution is `x = (c - b)/a`, which may be a non-integer or a fraction.
• When solving equations that include fractions, multiplying both sides by the lowest common multiple (LCM) of the denominators clears the fractions without changing the solution set (provided denominators are non-zero).
• Expanding brackets uses the distributive law, for example `k(x + m) = kx + km` and `(x + p)(x + q) = x^2 + (p + q)x + pq`.
• For equations with the unknown on both sides, collecting like terms gives a single `x`-term; if the `x`-terms cancel and constants match there are infinitely many solutions, and if they cancel but constants differ there is no solution.
• An inequality reverses its inequality sign when both sides are multiplied or divided by a negative number, for example from `<` to `>` and from `≥` to `≤`.
• Solutions to one-variable inequalities are represented on a number line with a solid dot for `≤` or `≥` and an open dot for `<` or `>`, with shading in the direction of the solution set.
• Changing the subject of a formula uses inverse operations to isolate the new subject; for example, from `y = 3x - 5` the subject `x` is `x = (y + 5)/3`.
• If the subject appears on both sides of a formula, terms containing the subject are brought to one side and then factored; for example, from `P = ax + b + cx` we obtain `x = (P - b)/(a + c)` provided `a + c ≠ 0`.
• A pair of simultaneous linear equations in two variables has a unique solution when their graphs are non-parallel straight lines that intersect at one point.
• Parallel but distinct lines have no solution to the simultaneous system, while coincident lines have infinitely many solutions; this is detectable by proportional coefficients with different or identical constants respectively.
• A monic quadratic `x^2 + bx + c = 0` factorises to `(x + p)(x + q) = 0` where `p + q = b` and `pq = c`, giving roots `x = -p` and `x = -q`.
• The quadratic formula gives the solutions of `ax^2 + bx + c = 0` as `x = (-b ± √(b^2 - 4ac)) / (2a)` for `a ≠ 0`.
• The discriminant `Δ = b^2 - 4ac` determines the nature of quadratic roots: `Δ > 0` gives two distinct real roots, `Δ = 0` gives one repeated real root, and `Δ < 0` gives no real roots.
• Completing the square rewrites `ax^2 + bx + c` (with `a ≠ 0`) as `a(x - h)^2 + k`, from which roots (if real) are found by solving `x - h = ±√(-k/a)`.
• When solving equations with algebraic fractions, any value of the variable that makes a denominator zero is excluded from the solution set (a restriction).
• To solve an equation with algebraic fractions, rearrange to a single rational expression and multiply through by the common denominator before solving the resulting polynomial equation.
• A linear inequality in two variables `ax + by ≤ c` represents a closed half-plane bounded by the line `ax + by = c` (solid line for `≤` or `≥`, dashed for `<` or `>`).
• A quadratic inequality in one variable is solved by finding the roots of the corresponding quadratic equation and using the parabola’s direction (sign of `a`) to determine intervals where the expression is positive or negative.
• An iterative process generates a sequence `x_{n+1} = g(x_n)` to approximate a solution; convergence near a fixed point is likely when `|g'(x)| < 1` in the neighbourhood.
• An approximate solution to an equation is stated to a specified degree of accuracy and is justified by showing that the function changes sign in a bounding interval or that successive iterates agree to the required precision.

• Solve a two-step linear equation `ax + b = c` by subtracting/adding `b` and then dividing/multiplying by `a`, giving the solution as an exact fraction when appropriate.
• Solve equations that include fractions and brackets by expanding brackets, identifying the LCM of denominators, multiplying through to clear fractions, simplifying, and checking that no denominator is zero at the solution.
• Solve linear equations with the unknown on both sides by collecting like terms, isolating the variable, and identifying special cases that yield no solution or infinitely many solutions.
• Solve two-step inequalities and represent the solution on a number line, correctly reversing the inequality when multiplying or dividing by a negative number.
• Change the subject of a formula in two steps using inverse operations, writing the subject explicitly; verify by substituting values to check equivalence.
• Rearrange a formula with the subject on both sides by bringing all subject terms to one side, factoring the subject, and dividing by the common factor to isolate it.
• Form and solve linear equations from worded contexts by defining a variable, translating conditions into an equation, solving accurately, and interpreting the solution in context.
• Solve linear simultaneous equations by elimination (matching coefficients, adding/subtracting to eliminate a variable) or substitution (substituting one equation into the other), stating the solution as an ordered pair and checking in both originals.
• Solve quadratic equations by factorising where possible, setting each factor to zero, and deducing exact roots; verify by substitution into the original equation.
• Solve quadratic equations using the quadratic formula or by completing the square, presenting exact surds when appropriate and decimal approximations to a stated accuracy, and commenting on the nature of roots using the discriminant when relevant.