National 5 Mathematics: relationships

Relationships is the part of National 5 Maths where algebra meets the coordinate grid: solving equations and inequations, handling two equations at once, finding the gradient and the equation of a straight line, and reading a quadratic for its roots, its turning point and how many times it meets the x-axis.

Ten single-answer questions work through exactly those. Two of them are the ones that cost marks most reliably: an inequation divided by a negative, where the sign has to be reversed, and a discriminant, where using the wrong sign inside the formula turns "no real roots" into "two real roots". The simultaneous-equations question is built so that each wrong pair satisfies one of the two equations but fails the other — guessing from a single line does not work. Explanations show the working line by line and name the slip that produces the closest wrong answer.

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• Solve a linear equation with a bracket on one side
• Solve an inequation and reverse the sign when dividing by a negative number
• Solve a pair of simultaneous equations by elimination and check both equations
• Find a gradient from two points, including negative coordinates
• Find the equation of a line from its gradient and a point on it
• Read the roots of a factorised quadratic with the correct signs
• State the turning point from vertex form and say whether it is a maximum or a minimum
• Use the discriminant to decide how many real roots an equation has
• Evaluate a function at a negative value
• Solve a quadratic by factorising when the leading coefficient is not 1

How many real roots does the quadratic equation $2x^2 - 5x + 4 = 0$ have? — The discriminant $D = b^2 - 4ac = (-5)^2 - 4(2)(4) = 25 - 32 = -7$. Since $D < 0$, there are no real roots.