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What is the difference between a linear equation and a cubic equation?
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What is the difference between a linear equation and a cubic equation?
Answer
<p>The difference between a linear equation and a cubic equation:</p> <p>Each equation gets its form based on the highest <span class="itxtnewhook itxthook" style="font-weight:normal;font-size:100%;font-style:normal;text-decoration:none;border:0px none transparent;padding:0px;background-color:transparent;background-image:none;"><span class="itxtrst itxtrstspan itxtnowrap">degree </span></span>or exponent, of the variable. For instance, in the case where y = x³ -- 6x + 2, the degree of 3 gives this equation the name "cubic." Any equation that has a degree no higher than 1 receives the name "linear." Otherwise, we call an equation "nonlinear," whether it is quadratic, a sine-curve or in any other form.</p> <p> </p> <div style="color:#000000;background-color:#ffffff;text-align:left;text-decoration:none;"><br /><br /></div>
Topics
Simplify expressions with surds into the form a + b3
Use index laws to evaluate any expression involving powers
Find the upper and lower bounds of calculations
Solve fractional equations with the unknown in the denominator
Simplify algebraic fractions
Solve quadratic equations by using the quadratic formula
Use the technique of completing the square
Solve a pair of simultaneous equations where one is linear and the other is a circle
Solve cubic equations by drawing appropriate lines on graphs
Transform the graphs of y = f(x), such as linear, quadratic, cubic, sine and cosine functions, using the transformations y = f(x) + a, y = f(x + a), y = f (ax) and y = af(x)
Recognise and sketch the shapes of graphs of functions
Derive algebraic proofs using reasoning and logic
Construct the graphs of loci, including the circle x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup>
Use trigonometry in three dimensions
Understand the graphs of trigonometric functions for angles of any size
Find the volume of the frustum of a truncated cone
Solve vector geometry problems
Solve probability problems involving dependent events
Rationalise the denominator of a surd
Use index laws to evaluate simple fractional or negative powers
Solve proportion problems (direct, inverse, proportional to the square, inversely proportional to the square)
Interpret the graphs of direct and inverse proportion relationships
Rearrange formulae where the new subject appears twice
Understand the gradients of perpendicular straight-line graphs
When given a graph, draw another line to solve an equation
Solve a pair of simultaneous equations where one is linear and the other involves x<sup>2</sup>
Calculate the length of an arc and the area of a sector and a segment of a circle
Calculate the surface areas of cylinders, cones and spheres
Calculate the volumes of cylinders, cones and spheres
Prove and use the circle theorems
Enlarge a shape by a negative scale factor
Solve problems involving the enlargement of areas and volumes
Solve vector geometry problems by adding, subtracting or multiplying with vectors
Use congruent triangles to solve problems and to provide proofs
Use Pythagoras theorem in 3-D problems
Use the cosine rule in any triangle
Use the sine rule in any triangle
Work out the area of a triangle using sin
Use stratified sampling
Interpret statistical graphs and draw conclusions
Construct and interpret a histogram
Solve probability problems involving independent events
Convert recurring decimals to fractions and fractions to recurring decimals
Identify recurring and terminating decimals
Calculate using standard index form
Convert between ordinary and standard index form representations
Work out reverse percentage problems
Work out compound interest
Calculate proportional changes using a multiplier
Solve quadratic equations by factorising
Factorise quadratic expressions
Expand and simplify quadratics
Rearrange formulae that include brackets, fractions and square roots
Use and understand y=mx+c
Solve a set of linear inequalities in 2 variables by shading a region of a graph
Solve a pair of simultaneous equations algebraically
Solve simultaneous equations graphically
Complete tables for, and draw graphs of cubic functions and quadratics
Calculate angles where circles are involved
Discuss and interpret graphs modelling real situations
Interpret velocity-time graphs
Solve problems involving similar triangles
Calculate the distance between two points using their coordinates
Use sin, cos and tan in a right-angled triangle
Compare two sets of data using averages
Construct and interpret a box plot
Construct and interpret a cumulative frequency diagram
Work out the median, quartiles and interquartile range
Find the least common multiple (LCM)
Find the highest common factor (HCF)
Write a number as a product of prime factors
Find the reciprocal of a number
Estimate answers to calculations
Use a calculator correctly and efficiently
Find the minimum and maximum values of a number that has been rounded off
Round a number off to a given number of significant figures
Understand the effects of multiplying by numbers between 0 and 1
Divide a number by a decimal
Work out a percentage increase or decrease
Express one quantity as a percentage of another
Solve ratio and proportion problems such as sharing out
Solve ratio and proportion problems using the unitary method
Calculate with fractions
Factorise expressions with one pair of brackets
Multiply out expressions with brackets
Use index notation and index laws for positive and negative powers
Write the nth term of a sequence or a series of diagrams
Find a solution to a problem by forming an equation and solving it
Solve linear equations
Form and solve equations such as x<sup>3</sup> + x = 12 using trial and improvement methods
Recognise the equations of straight-line graphs
Find the gradients of straight-line graphs
Draw graphs of quadratic functions by plotting points
Represent solutions to inequalities on a number line
Solve linear inequalities
Understand the difference between a demonstration and a proof
Convert between different units of area and volume
Calculate the volume of prisms and cylinders
Solve problems involving perimeters and areas of circles
Solve problems involving surface areas of prisms and cylinders
Classify a quadrilateral by its geometric properties
Solve problems using angle and symmetry properties of polygons
Understand and use coordinates in three dimensions
Find the midpoint of a line segment
Transform shapes by a combination of translation, reflection and rotation and know that area and angles are unchanged
Describe fully, and carry out reflections in lines and rotations about any point
Reflect shapes in the lines y = x and y = -x
Translate a shape by a vector
Enlarge a shape by any positive scale factor
Calculate average speeds from distance-time graphs
Understand and use compound measures such as speed and density
Prove that angles in a triangle add up to 180 degrees
Use Pythagoras theorem in practical problems
Use Pythagoras theorem
Solve loci problems
Construct an accurate locus
Carry out accurate constructions, using ruler and compass
Find averages from grouped data
Use a line of best fit to estimate values on a scatter graph and describe the correlation
Draw a line of best fit on the scatter graph
Identify possible sources of bias in the design and use of data collection sheets & questionnaires
Specify hypotheses and test them
Use relative frequency to find probabilities
Create a tree diagram
Understand and learn the words: square, positive and negative square root, cube and cube root, learn squares up to 15x15 and cubes up to 5 cubed
Multiply two numbers involving decimals
Increase or decrease a quantity by a given percentage
Convert between decimals, fractions and percentages
Recognise the words: equation, formula, identity and expression
Write the terms of a sequence or a series of diagrams when given the nth term
Draw lines such as y = 2x + 3 using a table of values
Solve equations with unknowns on each side and with brackets
Substitute numbers into formulas
Find the area and perimeter of compound shapes
Calculate the area of a circle
Calculate the circumference of a circle
Find the area of a triangle, parallelogram, kite and trapezium
Draw and interpret maps and scale drawings
Reflect shapes in lines such as x = 2 or y = -1
Identify reflection symmetry in 3-D shapes
Translate a shape using directions
Enlarge a shape by a positive integer scale factor
Solve problems using bearings
Solve problems using the properties of intersecting and parallel lines
Draw a quadrilateral such as a kite or a parallelogram with given measurements
Construct and recognise the nets of 3-D shapes such as pyramids and triangular prisms
Draw plans and elevations of 3-D shapes
Calculate the mean for a frequency distribution
Construct an ordered stem-and-leaf diagram
Construct a frequency diagram
Draw and interpret pie charts
Draw a scatter graph by plotting points on a graph and interpret it
Classify and know the difference between various types of data
Design and use data collection sheets and questionnaires
Use a variety of different sampling methods
Use a two-way table to work out probability
List all the possible outcomes of two events, using a probability space
Understand probabilities of mutually exclusive events