MathsQuestions

A Level
Quadratics

1. Factorise:

$x^2+7x+6$
$9x^2-16$

$(x+6)(x+1)$
$(3x+4)(3x-4)$

2. Factorise:

$4x^2-23x+15$
$-2x^2+9x-4$

$(4x-3)(x-5)$
$(1-2x)(x-4) = (2x-1)(4-x)$

3. A circle has equation $$2x^2+2y^2-16x+12y=0$$ Find the center and the radius of the circle.

Centre $(4,-3)$, radius 5

4. Solve the following by factorising:

$x^2-3x=10$
$8x-x^2=15$

$x = -2, 5$
$x = 3, 5$

5. Solve the following by completing the square:

$2(x^2-2x)=5$
$x(x-1)=3$

$1\pm\sqrt{\dfrac{7}{2}}$
$\dfrac{1}{2}\pm\dfrac{\sqrt{13}}{2}$

6. Solve the following using the quadratic equation:

$35=2x+x^2$
$2x^2+3=6x$

$x=-7,5$
$x=\dfrac{3\pm\sqrt{3}}{2}$

7. Two circles have equations: $$(x-3)^2+(y-2)^2 = 16$$ $$(x+5)^2+y^2=16$$ Determine if they intersect.

Distance between centres: $\sqrt{8^2+2^2}$
Sum of radii is smaller so no intersection.

8. Solve $$3x^2+(8-3x)^2 = 28$$

$x = 1, 3$

9. Sketch the following graphs, indicating clearly their points of intersection with the axes and the coordinates of their turning points:

$y=x^2-9$
$y=2x^2-5x+3$

$(-3,0)$, $(3,0)$, $(0, -9)$ minimum at $(0,9)$
$(1,0)$, $(1.5,0)$, $(0, 3)$ minimum at $(\frac{5}{4},-\frac{1}{8})$

10. A circle has centre $(-1,2)$ and passes through the points $A(-4,3)$ and $B(0,5)$.

Find an equation of the circle.
Find the equation of the perpendicular bisector of $AB$.

$(x+1)^2 + (y-2)^2 = 10$
$y = -2x$

11. A circle has a diameter between $(8,-7)$ and $(4,5)$. Find an equation of the circle.

$(x-6)^2 + (y+1)^2 = 40$

12. The function $\mathrm{f}(x) = x^2+3px+14p-3$, where $p$ is an integer, has two equal roots. Find $p$ and solve the equation for this value of $p$.

$p = 6$ and $x = -9$

13. By completing the square, find the turning points of:

$y=2x^2-x-5$
$y=2x+9-x^2$

$\left(\dfrac{1}{4},-\dfrac{41}{8}\right)$
$(1,10)$

14. Write the following in the form $a(x+p)^2+q$:

$2x^2+5x+1$
$7x-5x^2+11$

$2\left(x+\dfrac{5}{4}\right)^2-\dfrac{17}{8}$
$-5\left(x-\dfrac{7}{10}\right)^2+\dfrac{269}{20}$

15. A circle has equation $$x^2 + y^2 - 20x + 8y + 16 = 0$$

Find the gradient of the line segment joining the centre of the circle to the point $(4,4)$.
Hence find the equation of the tangent to the circle at the point $(4,4)$.

$-\dfrac{4}{3}$
$y=\dfrac{3}{4}x+1$

16. A circle has centre $(3,-5)$ and equation $x^2 + y^2 - 6x + ay = 15$, where $a$ is a constant. Find the radius of the circle.

$7$

17. A circle has diameter which has endpoints $(2,5)$ and $(-2,9)$.

Find an equation of the circle.
Does the point $(1,5)$ lie within the circle?

$x^2+(y-7)^2 = 8$
Yes

18. The points $P(-3,0)$, $Q(-1,6)$ and $R(11,2)$ lie on a circle.

Show that angle $PQR$ is a right angle.
Find an equation for the circle.

$PQ$ gradient $3$, $QR$ gradient $-\frac{1}{3}$
$(x-4)^2+(y-1)^2=50$

19. By finding the discriminant of $x^2-3x+5$, explain why $(x+2)(x^2-3x+5)$ is always positive for $x > -2$.

The quadratic has no real roots and is always positive. The linear part is also positive when $x > -2$.

20. Two circles have equations: $(x-7)^2+(y-3)^2=144$ and $(x+2)^2+(y-1)^2=9$. Determine if they intersect.

Distance between centres $\sqrt{9^2+2^2}$ is less than larger radius.
Difference between radii is smaller than distance between centres, so intersect at two points.

21. Find the minimum value of $$(x^2+4x+9)^2$$

$((x+2)^2+5)^2$ has minimum $25$

22. Given $\mathrm{f}(x) = 2x^2+(k+4)x+k$, where $k$ is a real constant, find the discriminant in terms of $k$ and hence prove that there are two distinct real roots of $\mathrm{f}(x)$ for all values of $k$.

$\Delta = (k+4)^2-8k = k^2+16$. Positive for all values of $k$, so there are always two real roots.

23. Solve the following:

$x^4-13x^2+36=0$
$x^6+7x^3=8$

$\pm 2, \pm 3$
$-2, 1$

24. Solve $$x+3=4\sqrt{x}$$

$1, 9$

25. Solve $$\dfrac{4x^4-24}{10} = x^2$$

$\pm 2$

26. Solve $$(x^2-4x+1)^2+(x^2-4x+1)=12$$

$2 \pm \sqrt{6}$

27. Solve $$2^{2x}+2^4=2^{x+1}+2^{x+3}$$

$1, 3$

28. Solve $$81 +3^{2x+1}=4\times 3^{x+2}$$

$1, 2$

29. A circle has centre at the origin and radius $r$. It fits completely inside another circle with equation $$x^2 + y^2 - 10x - 24y = 231$$ Find the range of possible values of $r$.

$r < 7$

30. The function $\mathrm{f}$ is such that $\mathrm{f}(x^n) = \mathrm{f}(x)^n$. Find the possible values of $\mathrm{f}(2)$ given $$2\mathrm{f}(64) - 7\mathrm{f}(16) = 4\mathrm{f}(4)$$

$-\frac{1}{2}, 0, 4$