GCSE Maths | Quadratics
Question 1
Factorise the following quadratics.
(a) $x^2+7x+12$
(b) $x^2+6x+8$
(c) $x^2+5x+6$
(d) $x^2+8x+7$
(e) $x^2+4x+4$
(f) $x^2+8x+15$
(g) $x^2+6x+9$
(h) $x^2+11x+28$
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(a) $(x+3)(x+4)$
(b) $(x+2)(x+4)$
(c) $(x+2)(x+3)$
(d) $(x+1)(x+7)$
(e) $(x+2)^2$
(f) $(x+3)(x+5)$
(g) $(x+3)^2$
(h) $(x+4)(x+7)$
Question 2
Factorise the following quadratics.
(a) $x^2+x-12$
(b) $x^2+5x-6$
(c) $x^2+3x-10$
(d) $x^2+3x-4$
(e) $x^2+2x-48$
(f) $x^2+4x-32$
(g) $x^2+2x-35$
(h) $x^2+8x-33$
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(a) $(x-3)(x+4)$
(b) $(x-1)(x+6)$
(c) $(x-2)(x+5)$
(d) $(x-1)(x+4)$
(e) $(x-6)(x+8)$
(f) $(x-4)(x+8)$
(g) $(x-5)(x+7)$
(h) $(x-3)(x+11)$
Question 3
Factorise the following quadratics.
(a) $x^2-3x-10$
(b) $x^2-x-20$
(c) $x^2-6x-27$
(d) $x^2-2x-3$
(e) $x^2-x-12$
(f) $x^2-4x-12$
(g) $x^2-4x-21$
(h) $x^2-6x-55$
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(a) $(x-5)(x+2)$
(b) $(x-5)(x+4)$
(c) $(x-9)(x+3)$
(d) $(x-3)(x+1)$
(e) $(x-4)(x+3)$
(f) $(x-6)(x+2)$
(g) $(x-7)(x+3)$
(h) $(x-11)(x+5)$
Question 4
Factorise the following quadratics.
(a) $x^2-6x+9$
(b) $x^2-9x+20$
(c) $x^2-9x+14$
(d) $x^2-13x+22$
(e) $x^2-9x+8$
(f) $x^2-12x+32$
(g) $x^2-15x+36$
(h) $x^2-14x+48$
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(a) $(x-3)^2$
(b) $(x-4)(x-5)$
(c) $(x-2)(x-7)$
(d) $(x-2)(x-11)$
(e) $(x-1)(x-8)$
(f) $(x-4)(x-8)$
(g) $(x-3)(x-12)$
(h) $(x-6)(x-8)$
Question 5
Factorise the following quadratics.
(a) $x^2-9x+8$
(b) $x^2+24x+23$
(c) $x^2-5x-14$
(d) $x^2-7x+12$
(e) $x^2+12x+36$
(f) $x^2-2x-63$
(g) $x^2+14x+24$
(h) $x^2+17x+60$
(i) $x^2-11x+30$
(j) $x^2-4x-32$
(k) $x^2-2x-63$
(l) $x^2-16x-17$
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(a) $(x-1)(x-8)$
(b) $(x+1)(x+23)$
(c) $(x-7)(x+2)$
(d) $(x-3)(x-4)$
(e) $(x+6)^2$
(f) $(x-9)(x+7)$
(g) $(x+2)(x+12)$
(h) $(x+5)(x+12)$
(i) $(x-5)(x-6)$
(j) $(x-8)(x+4)$
(k) $(x-9)(x+7)$
(l) $(x-17)(x+1)$
Question 6
Factorise the following quadratics using the difference of two squares.
(a) $x^2-25$
(b) $y^2-49$
(c) $w^2-100$
(d) $x^2-4$
(e) $c^2-64$
(f) $x^2-1$
(g) $x^2-900$
(h) $y^2-9$
(i) $16-x^2$
(j) $1-y^2$
(k) $81-x^2$
(l) $144-h^2$
(m) $x^2-y^2$
(n) $a^2-c^2$
(o) $9x^2-25$
(p) $4y^2-1$
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(a) $(x-5)(x+5)$
(b) $(y-7)(y+7)$
(c) $(w-10)(w+10)$
(d) $(x-2)(x+2)$
(e) $(c-8)(c+8)$
(f) $(x-1)(x+1)$
(g) $(x-30)(x+30)$
(h) $(y-3)(y+3)$
(i) $(4-x)(4+x)$
(j) $(1-y)(1+y)$
(k) $(9-x)(9+x)$
(l) $(12-h)(12+h)$
(m) $(x-y)(x+y)$
(n) $(a-c)(a+c)$
(o) $(3x-5)(3x+5)$
(p) $(2y-1)(2y+1)$
Question 7
Solve the following quadratics.
(a) $(x-1)(x-3)=0$
(b) $(y-4)(y-9)=0$
(c) $(m+1)(m+6)=0$
(d) $(x-3)(x+2)=0$
(e) $(t+7)(t-3)=0$
(f) $(k-10)(k+9)=0$
(g) $(w+5)(w+11)=0$
(h) $(y-8)(y-2)=0$
(i) $(x+3)(x-9)=0$
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(a) $x=1$ or $x=3$
(b) $y=4$ or $y=9$
(c) $m=-1$ or $m=-6$
(d) $x=3$ or $x=-2$
(e) $t=-7$ or $t=3$
(f) $k=10$ or $k=-9$
(g) $w=-5$ or $w=-11$
(h) $y=8$ or $y=2$
(i) $x=-3$ or $x=9$
Question 8
Solve the following quadratics using the quadratic formula.
(a) $x^2+5x+1=0$
(b) $2x^2+7x+2=0$
(c) $4x^2+8x+3=0$
(d) $x^2+2x-4=0$
(e) $3x^2+4x-5=0$
(f) $2x^2+5x-10=0$
(g) $x^2-4x+2=0$
(h) $7x^2-6x+1=0$
(i) $3x^2-10x+4=0$
(j) $x^2-x-11=0$
(k) $x^2-6x-20=0$
(l) $2x^2-x-9=0$
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(a) $x=\dfrac{-5\pm\sqrt{21}}{2}$
(b) $x=\dfrac{-7\pm\sqrt{33}}{4}$
(c) $x=-\dfrac{1}{2}$ or $x=-\dfrac{3}{2}$
(d) $x=-1\pm\sqrt{5}$
(e) $x=\dfrac{-2\pm\sqrt{19}}{3}$
(f) $x=\dfrac{-5\pm\sqrt{105}}{4}$
(g) $x=2\pm\sqrt{2}$
(h) $x=\dfrac{3\pm\sqrt{2}}{7}$
(i) $x=\dfrac{5\pm\sqrt{13}}{3}$
(j) $x=\dfrac{1\pm3\sqrt{5}}{2}$
(k) $x=3\pm\sqrt{29}$
(l) $x=\dfrac{1\pm\sqrt{73}}{4}$
Question 9
Complete the square in the following quadratics, then state the coordinates of the maximum or minimum point.
(a) $y=x^2-8x-2$
(b) $y=x^2+6x+10$
(c) $y=x^2-4x+1$
(d) $y=x^2+4x+9$
(e) $y=x^2+8x+20$
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(a) $(x-4)^2-18$; minimum point $(4,-18)$
(b) $(x+3)^2+1$; minimum point $(-3,1)$
(c) $(x-2)^2-3$; minimum point $(2,-3)$
(d) $(x+2)^2+5$; minimum point $(-2,5)$
(e) $(x+4)^2+4$; minimum point $(-4,4)$
Question 10
Complete the square in the following quadratics, then state the coordinates of the maximum or minimum point.
(a) $y=4x^2-8x+28$
(b) $y=5x^2+20x+15$
(c) $y=3x^2+24x+45$
(d) $y=4x^2-8x+16$
(e) $y=5x^2+10x$
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(a) $4(x-1)^2+24$; minimum point $(1,24)$
(b) $5(x+2)^2-5$; minimum point $(-2,-5)$
(c) $3(x+4)^2-3$; minimum point $(-4,-3)$
(d) $4(x-1)^2+12$; minimum point $(1,12)$
(e) $5(x+1)^2-5$; minimum point $(-1,-5)$
Question 11
Factorise the following quadratics.
(a) $2x^2+7x+5$
(b) $2x^2+11x+15$
(c) $2x^2+9x+10$
(d) $3x^2+13x+4$
(e) $3x^2+4x+1$
(f) $3x^2+8x+4$
(g) $5x^2+13x+6$
(h) $5x^2+26x+5$
(i) $7x^2+10x+3$
(j) $11x^2+47x+12$
(k) $2x^2+17x+36$
(l) $5x^2+62x+24$
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(a) $(x+1)(2x+5)$
(b) $(x+3)(2x+5)$
(c) $(x+2)(2x+5)$
(d) $(x+4)(3x+1)$
(e) $(x+1)(3x+1)$
(f) $(x+2)(3x+2)$
(g) $(x+2)(5x+3)$
(h) $(x+5)(5x+1)$
(i) $(x+1)(7x+3)$
(j) $(x+4)(11x+3)$
(k) $(x+4)(2x+9)$
(l) $(x+12)(5x+2)$
Question 12
Solve the following quadratic inequalities.
(a) $(x-4)(x-1)<0$
(b) $(x-2)(x+1)<0$
(c) $(x+7)(x+3)\leq 0$
(d) $(x-5)(x+4)\leq 0$
(e) $x(x-9)>0$
(f) $(x+6)(x-5)>0$
(g) $(x+10)(x+1)\geq 0$
(h) $(x-7)(x+7)\geq 0$
(i) $(x+8)(x+2)<0$
(j) $(x-4)(x+7)\geq 0$
(k) $(x+1)(x-5)\leq 0$
(l) $(x-12)(x-11)>0$
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(a) $1<x<4$
(b) $-1<x<2$
(c) $-7\leq x\leq -3$
(d) $-4\leq x\leq 5$
(e) $x<0$ or $x>9$
(f) $x<-6$ or $x>5$
(g) $x\leq -10$ or $x\geq -1$
(h) $x\leq -7$ or $x\geq 7$
(i) $-8<x<-2$
(j) $x\leq -7$ or $x\geq 4$
(k) $-1\leq x\leq 5$
(l) $x<11$ or $x>12$
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