How do different representations help us to understand quadratics?

Key questions

  1. 1

    How are the graphical and algebraic representations of quadratics connected?

  2. 2

    What do we mean by the roots of a quadratic equation and how can we find them?

  3. 3

    What is the discriminant of a quadratic, and what can it tell us?

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Resource type Title
Rich example Quadratic grids


Resource type Title
Building blocks Which quadratic?
Fluency exercise Quadratic solving sorter
Package of problems Discriminating
Package of problems Factorisable quadratics
Package of problems GeoGebra constructions... quadratic edition
Package of problems Irrational roots
Package of problems Name that graph
Package of problems Paired parabolas
Package of problems Quadratic symmetry
Package of problems Which parabola?
Scaffolded task Proving the quadratic formula
Food for thought Parabella
Food for thought Powerful quadratics
Bigger picture Parabolic mirrors
Resource in action Discriminating - teacher support

Review questions

Title Ref
Can we find both roots if one is double the other? R6638
Can we find the midpoint of a chord of this parabola? R8045
Can we find the tangents to a circle from a point? R6804
Can we fit a quadratic to this given data? R8775
Can we prove this inequality involving two square roots? R7533
Can we show $(b^2 - 2ac)x^2 + 4(a + c)x = 8$ always has real roots? R8812
Can we solve $12/(x-3) < x+1$? R7035
Can we solve $\sqrt{3-3x} - \sqrt{2-x} = 1$? R7485
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$? R6246
Can we solve the simultaneous equations $x + y + \sqrt{xy} = 39$ and $x^2 + y^2 + xy = 741$? R5682
Can we solve these simultaneous equations that involve reciprocals? R8128
Can we solve these simultaneous equations, both of degree two? R7520
Can we solve these simultaneous equations, one linear, one quadratic? R6429
Can we solve this inequality with three square roots? R6387
For what values of $k$ is $x^2 + 6kx +144$ always positive? R6526
Given the minimum point, what's this parabola's equation? R6756
Given two simultaneous equations of degree two, can we solve them? R7588
How do we solve $x + 3\sqrt{x} − 1/2 = 0$? R6215
How do we increase the roots of a quadratic equation by $1$? R6712
How do we solve an equation containing three square roots? R5847
If $2y = a^x + a^{-x}$, can we find $a^x$? R6816
If $7 - px - x^2 = 16 - (q + x)^2$, what are $p$ and $q$? R9724
If we know two values satisfying a quadratic, can we find the quadratic? R7972
Into how many regions do these parabolas divide the plane? R8781
What does the curve $x^4-y^2=y+1$ look like? R8855
When are the coefficients of a quadratic equal to its roots? R7350
When are the roots for $x^2-bx+c=0$ real and positive? R5138
When are these quadratic inequalities true together? R9989
When can this equation involving algebraic fractions hold? R8407
When do these simultaneous equations have no real solutions? R9658
When does $(p+1)x^2+4px+9=0$ have a repeated root? R7455
When does $x^2 + ax + a = 1$ have distinct real roots? R9659
When does $x^2+(3k-7)x+(2k+6)=0$ have real roots? R6828
When does $x^2-4x-1=2k(x-5)$ have equal roots? R5316
When does $x^4=(x-c)^2$ have four real roots? R9546
When does $y=kx$ intersect the parabola $y=(x-1)^2$? R9614
When does $y=mx-5$ intersect $y=x^2-1$ twice? R5287
When is $12x^2+7x-10$ negative? R5059
When is this line a tangent to this parabola? R7328
Where does $y = px^2 + 8x + p - 6$ cross the $x$-axis? R9742