A-Level Maths | Quadratics

Factorising Quadratics

Question 1

Factorise

(a) $x^2 + 4x + 3$

(b) $x^2 + 7x + 10$

(c) $y^2 - 3y + 2$

(d) $x^2 - 6x + 9$

(e) $y^2 - y - 2$

(f) $a^2 + 2a - 8$

(g) $x^2 - 1$

(h) $p^2 + 9p + 14$

(i) $x^2 - 2x - 15$

(j) $16 - 10m + m^2$

(k) $t^2 + 3t - 18$

(l) $y^2 - 13y + 40$

Completing the Square

Question 2

Express in the form $(x + a)^2 + b$

(a) $x^2 + 2x + 4$

(b) $x^2 - 2x + 4$

(c) $x^2 - 4x + 1$

(d) $x^2 + 6x$

(e) $x^2 + 4x + 8$

(f) $x^2 - 8x - 5$

(g) $x^2 + 12x + 30$

(h) $x^2 - 10x + 25$

(i) $x^2 + 6x - 9$

(j) $18 - 4x + x^2$

(k) $x^2 + 3x + 3$

(l) $x^2 + x - 1$

Question 3

Express in the form $a(x + b)^2 + c$

(a) $2x^2 + 4x + 3$

(b) $2x^2 - 8x - 7$

(c) $3 - 6x + 3x^2$

(d) $4x^2 + 24x + 1$

(e) $-x^2 - 2x - 5$

(f) $1 + 10x - x^2$

(g) $2x^2 + 2x - 1$

(h) $3x^2 - 9x + 5$

(i) $3x^2 - 24x + 48$

(j) $3x^2 - 15x$

(k) $70 + 40x + 5x^2$

(l) $2x^2 + 5x + 2$

Sketching Quadratics

Question 4

Sketch each curve showing the coordinates of any points of intersection with the coordinate axes.

(a) $y = x^2 - 3x + 2$

(b) $y = x^2 + 5x + 6$

(c) $y = x^2 - 9$

(d) $y = x^2 - 2x$

(e) $y = x^2 - 10x + 25$

(f) $y = 2x^2 - 14x + 20$

(g) $y = -x^2 + 5x - 4$

(h) $y = 2 + x - x^2$

(i) $y = 2x^2 - 3x + 1$

(j) $y = 2x^2 + 13x + 6$

(k) $y = 3 - 8x + 4x^2$

(l) $y = 2 + 7x - 4x^2$

(m) $y = 5x^2 - 17x + 6$

(n) $y = -6x^2 + 7x - 2$

(o) $y = 6x^2 + x - 5$

Question 5

Sketch each curve showing the exact coordinates of its turning point and the point where it crosses the $y$-axis.

(a) $y = x^2 - 4x + 3$

(b) $y = x^2 + 2x - 24$

(c) $y = x^2 - 2x + 5$

(d) $y = 30 + 8x + x^2$

(e) $y = x^2 + 2x + 1$

(f) $y = 8 + 2x - x^2$

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

(h) $y = -x^2 - 4x - 7$

(i) $y = x^2 - 5x + 4$

The Discriminant

Question 7

By evaluating the discriminant, determine whether the roots of each equation are real and distinct, real and equal, or not real.

(a) $x^2 + 2x - 7 = 0$

(b) $x^2 + x + 3 = 0$

(c) $x^2 - 4x + 5 = 0$

(d) $x^2 - 6x + 3 = 0$

(e) $x^2 + 14x + 49 = 0$

(f) $x^2 - 9x + 17 = 0$

(g) $x^2 + 3x = 11$

(h) $2 + 3x + 2x^2 = 0$

(i) $5x^2 + 8x + 3 = 0$

(j) $3x^2 - 7x + 5 = 0$

(k) $9x^2 - 12x + 4 = 0$

(l) $13x^2 + 19x + 7 = 0$

Question 8

The quadratic equation $x^2 + 10x + k = 0$, where $k$ is a constant, has no real roots.

Find the range of possible values of $k$.

Question 9

$f(x) \equiv 25x^2 + 20x + p$, where $p$ is a non-zero constant.

The quadratic equation $f(x) = 0$ has equal roots.

Find the value of $p$.

Question 10

The quadratic equation $mx^2 + 12x + m = 0$, where $m$ is a constant, has repeated roots.

Find the possible values of $m$.

Question 11

Find the range of values of the non-zero constant $k$, given that the quadratic equation $3kx^2 - 2kx - 4x + 3 = 0$ has two different real roots.

Question 12

It is given that $f(x) = x^2 + 2x - m(x^2 - 2x + 2) - 2$, where $m$ is a constant such that $m \neq 1$.

The equation $f(x) = 0$ has distinct real roots.

Determine the range of values of $m$.