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105 results.

Integration by partial fractions
Ready to use
Question in Bill's workspace by Bill Foster and 1 other

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.
Numbas demo: video
Ready to use
Question in Demos by Christian Lawson-Perfect

Customised for the Numbas demo exam

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Video in Show steps.
Logarithms: Solving equations 6
Draft
Question in Bill's workspace by Bill Foster

Solve for $x$ : $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).
Hyperbolic Functions 1
Draft
Question in Bill's workspace by Bill Foster

Solve for $x$ : $a\cosh(x)+b\sinh(x)=c$ . There are two solutions for this example.
Solving equations
Draft
Question in Bill's workspace by Bill Foster

Solve for $x$ : $\displaystyle ax ^ 2 + bx + c=0$ .
Expansion of two brackets: Linear and Quadratic
Draft
Question in Bill's workspace by Bill Foster

Expand $(pw+q)(aw^2+bw+c)$ .
Expansion of two brackets: Quadratic and Linear
Draft
Question in Bill's workspace by Bill Foster

Expand $(az^2+bz+c)(pz+q)$ .
Expansion of two brackets: Quadratic and Quadratic
Draft
Question in Bill's workspace by Bill Foster

Expand $(az^2+bz+c)(dz^2+pz+q)$ .
Complete the square. (Video)
Draft
Question in Bill's workspace by Bill Foster

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$ .

Hence, or otherwise, find roots of $ax^2+bx+c=0$ .

Includes a video which shows how to solve a quadratic by completing the square.
Combining algebraic fractions 6.0
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{a}{x + b} + \frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator.
Factorise a quadratic
Draft
Question in Bill's workspace by Bill Foster

Factorise $\displaystyle{ax ^ 2 + bx + c}$ into linear factors.
Complete the square
Draft
Question in Bill's workspace by Bill Foster

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$ .
Compute Riemann sums of a quadratic function
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.
Approximate the integral of a quadratic by Riemann sums,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.
Integration by partial fractions
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.
Integration by partial fractions
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.
Complete the square
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $B$ and $C$ such that $x^2+bx+c = (x+B)^2+C$ .
Complete the square
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $a$ , $B$ and $C$ such that $ax^2+bx+c = a(x+B)^2+C$ .
Intersection of a straight line and a circle
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the points of intersection of a straight line and a circle.
Intersection of two circles in two points
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find the points of intersection of two circles.
Second order ODE with constant coefficients and boundary conditions
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$ .
Solve a second order ODE with repeated roots,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$ . (Equal roots example).
Complete the square
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$ . Find both roots of the equation $q(x)=0$ .
Completing the square
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$ .
Solving equations
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle ax ^ 2 + bx + c=0$ .
Solve a quadratic with the formula
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 2 others

Solve for $x$ : $\displaystyle ax ^ 2 + bx + c=0$ .

Entering the correct roots in any order is marked as correct. However, entering one correct and the other incorrect gives feedback stating that both are incorrect.
Divide two elements of $\mathbb{Z}[\sqrt{2}]$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

In the ring $\mathbb{Z}[\sqrt{2}]$ , find the remainder $r=r_1+r_2\sqrt{2}$ , where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{2}$ by $c+d\sqrt{2}$ .
Use the quadratic formula to solve an equation in terms of an unknown variable
Ready to use
Question in Transition to university by Hannah Aldous and 2 others

Factorise a quadratic expression of the form $x^2+akx+bk^2$ for $x$ , in terms of $k$ . $a$ and $b$ are constants.
Solving Quadratic Equations with $x^2$ Coefficients of 1 by Factorising
Should not be used
Question in Transition to university by Hannah Aldous

This question tests the students ability to factorise simple quadratic equations (where the coefficient of the x^2 term is 1) and use the factorised equation to solve the equation when it is equal to 0.
Solve quadratic inequalities
Ready to use
Question in Transition to university by Bradley Bush and 1 other

This question takes the student through variety of examples of quadratic inequalities by asking them for the range(s) for which $x$ meets the inequality.

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