By completing the square, sketch the graph $y = x^2 + 2x + 4$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$(x+1)^2 + 3$, turning point: $(-1,3)$, $y$-intercept: $(0,4)$
By completing the square, sketch the graph $y = x^2 - 7x - 2$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$\left(x - \dfrac{7}{2}\right)^2 - \dfrac{57}{4}$, turning point: $\left(-\dfrac{7}{2}, -\dfrac{57}{4}\right)$, $y$-intercept: $(0,-2)$
By completing the square, sketch the graph $y = 2x^2 + 4x + 3$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$2\left(x + 1\right)^2 + 1$, turning point: $\left(-1, 1\right)$, $y$-intercept: $(0,3)$
By completing the square, sketch the graph $y = 4x^2 + 24x + 11$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$4\left(x + 3\right)^2 - 25$, turning point: $\left(-3, -25\right)$, $y$-intercept: $(0,11)$
By completing the square, sketch the graph $y = -x^2 - 2x - 5$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$-\left(x + 1\right)^2 - 4$, turning point: $\left(-1, -4\right)$, $y$-intercept: $(0,-5)$
By completing the square, sketch the graph $y = 1 + 10x - x^2$, showing the coordinates of the turning point and the intersection with the $y$ axis.
$-\left(x - 5\right)^2 + 26$, turning point: $\left(5, 26\right)$, $y$-intercept: $(0,1)$
By completing the square, find the equation of the line of symmetry of the graph of $y = 2x^2 + 4x - 1$.
$2\left(x + 1\right)^2 - 3$, line $x = -1$
By completing the square, find the equation of the line of symmetry of the graph of $y = 4 - 2x - 3x^2$.
$-3\left(x + \dfrac{1}{3}\right)^2 + \dfrac{13}{3}$, line $x = -\dfrac{1}{3}$
By completing the square, find the minimum value of $(x^2 + 10x + 35)^2$ and state the value of $x$ for which this occurs.
min: $100$ at $x = -5$
By completing the square, find the minimum value of $4x^2 - 12x + 9$ and state the value of $x$ for which this occurs.
min: $0$ at $x = \dfrac{3}{2}$
By first completing the square, solve $x^2 + 7x = 44$
$x = -\dfrac{7}{2} \pm \dfrac{15}{2}$
By first completing the square, solve $2x^2 - 4x + 1 = 0$
$x = 1 \pm \dfrac{\sqrt{2}}{2}$
By first completing the square, solve $3x^2 + 18x + 23 = 0$
$x = -3 \pm \dfrac{2\sqrt{3}}{3}$
By first completing the square, solve $-x^2 + x + 1 = 0$
$x = \dfrac{1}{2} \pm \dfrac{\sqrt{5}}{2}$
By first completing the square, solve $4x^2 + 49 = 28x$
$x = \dfrac{7}{2}$
By first completing the square, solve $1 - x = 3x^2$
$x = -\dfrac{1}{6} \pm \dfrac{\sqrt{13}}{6}$
A curve has equation $y = x^2 + bx + c$ and $(-3, 0)$ is the minimum point. Find the equation of the curve.
$x^2 + 6x + 9$
A curve has equation $y = x^2 + bx + c$ and $(0, 2)$ is the minimum point. Find the equation of the curve.
$x^2 + 2$
A curve has equation $y = x^2 + bx + c$ and $(2, 1)$ is the minimum point. Find the equation of the curve.
$x^2 - 4x + 5$
A curve has equation $y = x^2 + bx + c$ and $(5, -2)$ is the minimum point. Find the equation of the curve.
$x^2 - 10x + 23$
A curve has equation $y = ax^2 + bx + c$. $(2, 3)$ is the minimum point and it goes through the point $(1,5)$. Find the equation of the curve.
$2x^2 - 8x + 11$
A curve has equation $y = ax^2 + bx + c$. $(-3, 5)$ is the maximum point and it goes through the point $(-1,-7)$. Find the equation of the curve.
$-3x^2 - 18x - 22$
Write $x^2 - 4\sqrt{2}x + 5$ in the form $(x+p)^2 + q$ and hence write down the coordinates of the minimum point of the graph with equation $y = x^2 - 4\sqrt{2}x + 5$
$(x - 2\sqrt{2})^2 - 3$, minimum $(2\sqrt{2}, -3)$
Write $x^2 + 2kx - 3$ in the form $(x+p)^2 + q$ and hence solve the equation $x^2 + 2kx - 3 = 0$, leaving your answers in terms of $k$.
$(x + k)^2 - k^2 - 3$, $x = -k \pm \sqrt{k^2 + 3}$