@?maths

What does the set ${x \colon x < 6.5}$ describe?? All numbers that are less than $6.5$. What does the set ${x \colon x > -2}$ describe?? All numbers that are greater than $-2$. How could you write the set for all numbers less than or equal to $5$?? $$ { x \colon x \le 5 } $$ How could you describe $-2 \le x < 6.5$ as one set?? $$ { x \colon -2 \le x < 6.

Where does the value of $b^2 - 4ac$ come from for the discriminant?? The value under the square root sign in the quadratic formula. Why is the discriminant useful?? It tells you how many real roots a quadratic has. What is the formula for the discriminant?? $$ b^2 - 4ac $$ Why must a positive discriminant mean two real roots?? The square root of a positive number has two possible solutions. Why must a zero discriminant mean one repeated root?

See [[Further Maths - Solving Systems of Equations Using Triangle Method]]S [[Further Maths - Solving Systems of Equations Using Matricies]]S Backlinks [[Maths - Syllabus]]S Metadata date: 2020-10-24 13:06 tags: - '@?maths' - '@?public' title: Maths - Simultaneous Equations

Completing the Square (blank so that the linker doesn’t complain). Backlinks [[Maths - Syllabus]]S Metadata date: 2020-10-14 10:45 tags: - '@?maths' - '@?school' - '@?public' - '@?year-1' title: Maths - Completing the Square

What is a quartic?? A 4-th degree polynomial. What’s the largest number of (real) solutions a quartic can have?? $4$. What’s the smallest number of (real) solutions a quartic can have?? $0$. I want to make a quartic with only two roots, $3$ and $4$. How could I do this?? $$ (x-3)(x-3)(x-4)(x-4) $$ Visualise the quartic $x^2(x-3)(x+3)$?? PHOTO When sketching a quatic, what’s the process?? Add the roots to an axis Add the y-intercept Backlinks [[Maths - Syllabus]]S Metadata date: 2020-10-11 11:51 tags: - '@?

See Also https://www.desmos.com/calculator/oyp1sie3gb [[Maths - Modelling with Differentiation]]S [[Maths - Sketching Gradient Functions]]S [[Maths - Chain Rule]]S [[Maths - Product Rule]]S Flashcards What is the derivative of $x^n$?? $$ nx^{(n-1)} $$ What is the derivative of $ax^n$?? $$ anx^{(n-1)} $$ What is $f'(x)$ where $f(x) = 4x^2$?? $$ 8x $$ What is $\frac{dy}{dx}$ for $y = \frac{1}{2} x^{-4}$?? $$ -2x^{-5} $$ What is the derivative of $x^{\frac{1}{2}}$?

See Also [[Maths - Differentiation]]S [[Further Maths - Volumes of Revolutions]]S Flashcards What is $x^n$ integrated?? $$ \frac{x^{n+1}}{n+1} + c $$ What is $kx^n$ integrated?? $$ \frac{kx^{n+1}}{n+1} + c $$ Integrate $2x+4$?? $$ x^2 + 4x + c $$ What does $c$ mean in integration?? The constant of integration. Why is a constant of integration important?? Because it represents any constant term that would disappear when being differentiated. What does it mean to find a particular solution of an integral?

If the coefficient of $x^3$ is positive, how does a cubic graph look?? PHOTO If the coefficient of $x^3$ is negative, how does a cubic graph look?? PHOTO A cubic equation has roots $-1, 2$ and $3$. Visualise it’s graph?? PHOTO What does a graph of $\frac{1}{x}$ look like?? PHOTO What does a graph of $\frac{1}{x^2}$ look like?? PHOTO What does a graph of $\frac{-2}{x}$ look like?? PHOTO What does a graph of $\frac{-5}{x^2}$ look like?

How can you solve something like $-x^2 + 7x - 15$?? Take out $-1$ as a factor. What is the result of taking out $-1$ from $15 + 13x - 4x^2$?? $$ -(4x^2 - 13x - 15) $$ How can you solve something like $x^4 + 5x^2 - 36 = 0$?? Substitute $a = x^2$. Substitute $a = x^2$ into $x^4 + 5x^2 - 36 = 0$?? $$ a^2 + 5a - 35 = 0 $$

See Also [[Maths - Proof by Contradiction]]S Flashcards When proving irrationality of a surd $x$, what is the setup?? Assume $x = \frac{a}{b}$ where $a$ and $b$ are integers that do not share any factors. Why can’t $a$ and $b$ share any factors when proving the irrationality of a surd?? Because it would mean you could simplify $\frac{a}{b}$. How do you get rid the square root over $x$ when proving it’s irrationality?

What are integers?? All numbers without a fractional component. What are numbers without a fractional component called?? Integers. What are rational numbers?? Numbers that can be written in the form $\frac{a}{b}$. What are numbers that can be written as $\frac{a}{b}$ called??' Rational numbers. What are irrational numbers?? Non-recurring decimals. What are non-recurring decimals called?? Irrational numbers. What are real numbers?? Rational and irrational numbers. What the term for rational and irrational numbers?? Real numbers. Backlinks [[Maths - Syllabus]]S Metadata date: 2020-09-23 17:28 tags: - '@?

How do you find the intersection for something like $y = 5 - 2x$ and $y = 3x^2 + 5x - 3$?? Set them equal to each other and simplify. $5 - 2x = 3x^2 + 5x - 3$ What’s the most amount of times a straight line and a quadratic could intersect?? What’s the least amoount of times a straight line and a quadratic could intersect?? A line of the form $y = mx+c$ with one point of intersection with a quadratic has what relationship?