A-Level Maths | Quadratics
Question 1
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) $y=6x^2+x-5$
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(a) $(1,0)$, $(2,0)$, $(0,2)$
(b) $(-3,0)$, $(-2,0)$, $(0,6)$
(c) $(-3,0)$, $(3,0)$, $(0,-9)$
(d) $(0,0)$, $(2,0)$
(e) $(5,0)$, $(0,25)$
(f) $(2,0)$, $(5,0)$, $(0,20)$
(g) $(1,0)$, $(4,0)$, $(0,-4)$
(h) $(-1,0)$, $(2,0)$, $(0,2)$
(i) $\left(\frac{1}{2},0\right)$, $(1,0)$, $(0,1)$
(j) $(-6,0)$, $\left(-\frac{1}{2},0\right)$, $(0,6)$
(k) $\left(\frac{1}{2},0\right)$, $\left(\frac{3}{2},0\right)$, $(0,3)$
(l) $\left(-\frac{1}{4},0\right)$, $(2,0)$, $(0,2)$
(m) $\left(\frac{2}{5},0\right)$, $(3,0)$, $(0,6)$
(n) $\left(\frac{1}{2},0\right)$, $\left(\frac{2}{3},0\right)$, $(0,-2)$
(o) $(-1,0)$, $\left(\frac{5}{6},0\right)$, $(0,-5)$
Question 2
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) $y=x^2-5x+4$
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(a) $(2,-1)$, $(0,3)$
(b) $(-1,-25)$, $(0,-24)$
(c) $(1,4)$, $(0,5)$
(d) $(-4,14)$, $(0,30)$
(e) $(-1,0)$, $(0,1)$
(f) $(1,9)$, $(0,8)$
(g) $(4,9)$, $(0,-7)$
(h) $(-2,-3)$, $(0,-7)$
(i) $\left(\frac{5}{2},-\frac{9}{4}\right)$, $(0,4)$
Question 3
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) $13x^2+19x+7=0$
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(a) $\Delta=32$ — real and distinct
(b) $\Delta=-11$ — not real
(c) $\Delta=-4$ — not real
(d) $\Delta=24$ — real and distinct
(e) $\Delta=0$ — real and equal
(f) $\Delta=13$ — real and distinct
(g) $\Delta=53$ — real and distinct
(h) $\Delta=-7$ — not real
(i) $\Delta=4$ — real and distinct
(j) $\Delta=-11$ — not real
(k) $\Delta=0$ — real and equal
(l) $\Delta=-3$ — not real
Question 4
The quadratic equation $x^2+10x+k=0$, where $k$ is a constant, has no real roots.
Find the range of possible values of $k$.
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$k>25$
Question 5
$f(x)\equiv 25x^2+20x+p$, where $p$ is a non-zero constant.
The quadratic equation $f(x)=0$ has equal roots.
Find the value of $p$.
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$p=4$
Question 6
The quadratic equation $mx^2+12x+m=0$, where $m$ is a constant, has repeated roots.
Find the possible values of $m$.
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$m=\pm 6$
Question 7
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.
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$k<1$ or $k>4$, $k\neq 0$
Question 8
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 $m$.
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$-1<m<3$, $m\neq 1$
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