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neurologyYou need to know

• A linear equation is a statement of equality that can be solved by finding values that make both sides equal; an inequality uses the symbols <, ≤, >, or ≥ to describe ranges of values that satisfy the relation.
• When solving linear inequalities, multiplying or dividing both sides by a negative number reverses the inequality sign.
• On a number line, an open circle shows a strict inequality (< or >) and a closed circle shows an inclusive inequality (≤ or ≥); arrows indicate solutions extending to infinity.
• A two-step linear equation or inequality can be solved by reversing addition/subtraction and multiplication/division in the correct order while maintaining balance on both sides.
• Equations with the unknown on both sides are solved by collecting like terms, moving all variable terms to one side and constants to the other before isolating the variable.
• Changing the subject of a formula uses inverse operations applied to both sides while preserving equivalence; if the subject appears on both sides, terms are collected and the subject is factored out before dividing.
• For any quadratic equation `ax^2+bx+c=0` with `a≠0`, factorising to `(x-r_1)(x-r_2)=0` shows the roots are `x=r_1` and `x=r_2`.
• The quadratic formula gives the solutions to `ax^2+bx+c=0` as `x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}` provided `a≠0`.
• The discriminant `\Delta=b^2-4ac` determines the number of real roots of a quadratic: `\Delta>0` gives two distinct real roots, `\Delta=0` gives one repeated real root, and `\Delta<0` gives no real roots.
• Completing the square rewrites `ax^2+bx+c` (with `a≠0`) in the form `a(x+\frac{b}{2a})^2 - \frac{b^2-4ac}{4a}`; for `a=1`, `x^2+bx` becomes `(x+\tfrac{b}{2})^2-\tfrac{b^2}{4}`.
• Solving a quadratic by completing the square leads to an equation of the form `(x+p)^2=q`, whose solutions are `x=-p\pm\sqrt{q}` when `q≥0`.
• Simultaneous linear equations represent two lines; their solution is the ordered pair `(x,y)` at their intersection, which may be a unique point, no solution (parallel lines), or infinitely many solutions (coincident lines).
• A linear–quadratic simultaneous system represents the intersection(s) of a line and a parabola and can yield 0, 1, or 2 real solutions depending on whether the line cuts, touches, or misses the curve.
• In equations involving algebraic fractions, any value that makes a denominator zero is excluded from the solution set (domain restriction), and cross-multiplication is valid only when all denominators are non-zero.
• Clearing denominators in algebraic fraction equations by multiplying through by the lowest common multiple of denominators can create extraneous solutions, which must be checked against domain restrictions.
• A linear inequality in two variables represents a half-plane; a solid boundary line is used for ≤ or ≥, a dashed boundary line for < or >, and a test point can determine the correct region to shade.
• A quadratic inequality in one variable can be solved by finding the roots of the related quadratic equation and using a sign diagram (or the shape of the parabola) to determine intervals where the expression is positive or negative.
• An iterative process forms a sequence `x_{n+1}=f(x_n)` to approach a root; stable iterations show successive terms settling towards a value, which is then taken as an approximate solution to stated accuracy.

• Solve a two-step linear equation by undoing operations in reverse order, showing each inverse operation on both sides and verifying the solution by substitution.
• Solve equations with the unknown on both sides by collecting variable terms on one side, constants on the other, isolating the variable, and checking the solution.
• Change the subject of a two-step formula by applying inverse operations and, when the subject appears on both sides, factorising the subject before dividing to isolate it.
• Form and solve two-step equations from a worded context by translating the relationships into algebra, solving, and interpreting the solution with appropriate units.
• Represent solutions to one-variable inequalities on a number line using open/closed circles and arrows, matching the inequality symbols precisely.
• Solve linear simultaneous equations by elimination (matching coefficients, adding/subtracting to eliminate one variable) or substitution, and present the solution as an ordered pair.
• Solve quadratic equations by factorising into linear factors where possible and deducing the roots directly from the zero-product property.
• Use the quadratic formula accurately, including calculating the discriminant and evaluating surds, and state both roots (or the repeated root) to a specified degree of accuracy.
• Complete the square to solve a quadratic and to deduce roots, writing the quadratic in vertex form and solving `(x+p)^2=q` when `q≥0`.
• Solve equations involving algebraic fractions by stating domain restrictions, clearing denominators using the lowest common multiple, solving the resulting equation, and rejecting any values that invalidate a denominator or arise extraneously. Solve linear–quadratic simultaneous equations by substituting the linear expression into the quadratic, solving the resulting quadratic, and pairing each valid x-value with its corresponding y-value. Solve linear inequalities in two variables by drawing the boundary line, choosing solid or dashed styling, testing a point, and shading the correct half-plane. Solve quadratic inequalities in one variable by factorising, identifying critical values (roots), using a sign chart or parabola shape to select solution intervals, and expressing the final solution using inequality or interval notation as required. Find approximate solutions to equations using iteration by applying a given recurrence `x_{n+1}=f(x_n)` with a sensible starting value, iterating to the required accuracy, and stating the answer with correct rounding and error bounds where requested.