###### 1.1

Express $$5x^2 + 10x + 2$$ in the form $$p(x + q)^2 + r$$, where $$p$$$$q$$ and $$r$$ are integers.

$$5x^2 + 10x + 2 = 5(x^2 + 2x) + 2 = 5(x^2 + 2x + 1 - 1) + 2 = 5(x^2 + 2x + 1) - 5 + 2 = 5(x + 1)^2 - 3$$

###### 1.2

Given the quadratic equation $$3x^2 + 9x + 10$$.

Express $$3x^2 + 9x + 10$$ in the form $$3(x + 1.5)^2 + q$$ and find $$q$$.

$$3x^2 + 9x + 10 = 3(x^2 + 3x) + 10 = 3(x^2 + 3x + (1.5)^2 - (1.5)^2) + 10 = 3(x^2 + 3x + 2.25) - 5*2.25 + 10 = 3(x + 1.5)^2 - 11.25 + 10 = 3(x + 1.5)^2 - 1.25$$

###### 1.3

Solve the equation $$x^2 - 6x - 2 = 0$$, giving your answer in the form $$p \pm q\sqrt{11}$$.

$$D = b^2 - 4ac = (-6)^2 - 4\times(-2) = 36 + 8 = 44$$. $$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{2} = \frac{6 \pm 2\sqrt{11}}{2} = 3 \pm \sqrt{11}$$.

###### 1.4

Given the quadratic equation $$2x^2 - 20x + 49$$.

Express it in the form $$p(x + q)^2 + r$$.

$$2x^2 - 20x + 49 = 2(x^2 - 10x) + 49 = 2(x^2 - 10x + 25 - 25) + 49 = 2(x^2 - 10x + 25) - 50 + 49 = 2(x - 5)^2 + 1$$

###### 1.5

Given that $$5x^2 + px - 8 = q(x - 1)^2 + r$$. for all values of $$x$$.

Find the values of the constants $$p$$$$q$$ and $$r$$.

Open brackets on the right side of the equation: $$q(x - 1)^2 + r = q(x^2 - 2x + 1) + r = qx^2 -2qx + q + r$$. Since this expression must be equal to $$5x^2 + px - 8$$ for all values of $$x$$

\left\{ \begin{aligned} &q = 5\\ &p = -q\\ &q + r = -8 \end{aligned} \right.

Then,

\left\{ \begin{aligned} &p = -10\\ &q = 5\\ &r = -13 \end{aligned} \right.

###### 1.6

Express $$3x^2 - 18x + 4$$ in the form $$p(x + q)^2 + r$$.

Find $$r$$.

$$3x^2 - 18x + 4 = 3(x^2 - 6x) + 4 = 3(x^2 - 6x + 9 - 9) + 4 = 3(x^2 - 6x + 9) - 27 + 4 = 3(x - 3)^2 - 23$$

###### 1.7

Given the quadratic equation $$4x^2 + 12x - 3$$.

Express it in the form $$p(x + q)^2 + r$$. Find $$r$$.

$$4x^2 + 12x - 3 = 4(x^2 + 3x) - 3 = 4(x^2 + 3x + (1.5)^2 - (1.5)^2) - 3 = 4(x^2 + 3x + 2.25) - 4*2.25 - 3 = 4(x + 1.5)^2 - 9 - 3 = 4(x + 1.5)^2 - 12$$

###### 1.8

Given the quadratic equation $$2x^2 + 5x = 0$$.

Express $$2x^2 + 5x$$ in the form $$2(x + p)^2 + q$$.

$$2x^2 + 5x = 2(x^2 + 2.5x) = 2(x^2 + 2.5x + (1.25)^2 - (1.25)^2) = 2(x^2 + 2.5x + (1.25)^2) - 2\times1.5625 = 2(x^2 + 2.5x + (1.25)^2) - 3.125 = 2(x + 1.25)^2 - 3.125.$$

###### 1.9

Express $$x^2 − 12x + 1$$ in the form $$(x − p)^2 + q$$.

Find $$q$$.

$$x^2 − 12x + 1 = x^2 - 12x + 36 - 36 + 1 = (x - 6)^2 - 35$$

###### 1.10

Solve the equation $$9x^2 + 18x − 7 = 0$$.

Choose the roots of the equation.

$$D = b^2 - 4ac = (18)^2 - 4\times9\times(-7) = 324 + 252 = 576$$. $$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-18 \pm \sqrt{576}}{18} = \frac{-18 \pm 24}{18} = -1 \pm 0.75$$. Thus, $$x = -1.75$$ or $$x = 0.25$$.

###### 1

Given the quadratic equation $$3x^2 + 9x + 10$$.

State the coordinates of the minimum point of the curve $$y = 3x^2 + 9x + 10$$.

Since $$3x^2 + 9x + 10 = 3(x + 1.5)^2 - 1.25$$, the minimum value of the function $$y(x) = 3x^2 + 9x + 10$$ achieved at the point $$x = -1.5$$. The minimum value of the $$y(x)$$ is $$y(-1.5) = 3(-1.5 + 1.5)^2 - 1.25 = -1.25$$.

###### 2

Given the quadratic equation $$2x^2 - 20x + 49$$.

State the coordinates of the vertex of the curve $$y = 2x^2 - 20x + 49$$.

The vertex of the curve $$y = 2x^2 - 20x + 49$$ is the minimum value of the function $$y(x) = 2x^2 - 20x + 49$$. Since $$2x^2 - 20x + 49 = 2(x - 5)^2 + 1$$, the minimum value of the function $$y(x) = 2x^2 - 20x + 49$$ achieved at the point $$x = 5$$. The minimum value of the $$y(x)$$ is $$y(5) = 2(5 - 5)^2 + 1 = 1$$.

###### 3

Given the quadratic equation $$4x^2 + 12x - 3$$.

Solve the equation $$4x^2 + 12x - 3 = 0$$, giving your answer in the form $$p \pm q\sqrt{3}$$

Since $$4x^2 + 12x - 3 = 4(x + 1.5)^2 - 12 = 4((x + 1.5)^2 - 3) = 4(x + 1.5 - \sqrt{3})(x + 1.5 + \sqrt{3})$$, the equation $$4x^2 + 12x - 3 = 0$$ equivalent to $$x + 1.5 - \sqrt{3} = 0$$ or $$x + 1.5 + \sqrt{3} = 0$$. Thus, $$x = -1.5 + \sqrt{3}$$ or $$x = -1.5 - \sqrt{3}$$.

###### 4

Given the quadratic equation $$4x^2 + 12x - 3$$.

The quadratic equation $$4x^2 + 12x - k = 0$$ has equal roots. Find the value of $$k$$.

$$4x^2 + 12x - k = 4(x^2 + 3x) - k = 4(x^2 + 3x + (1.5)^2 - (1.5)^2) - k = 4(x^2 + 3x + 2.25) - 4*2.25 - k = 4(x + 1.5)^2 - 9 - k$$. Since the equation $$4x^2 + 12x - k = 0$$ has equal roots, it must have the form $$4(x + 1.5)^2 = 0$$, so $$-9 - k = 0$$. Thus, $$k = -9$$.

###### 5

Given the quadratic equation $$2x^2 + 5x = 0$$.

State the minimum value of the function $$y = 2x^2 + 5x$$.

Since $$2x^2 + 5x = 2(x + 1.25)^2 - 3.125$$, the minimum value of the function $$y(x) = 2x^2 + 5x$$ achieved at the point $$x = -1.25$$. The minimum value of $$y(x)$$ is $$y(-1.25) = 2(-1.25 + 1.25)^2 - 3.125 = -3.125$$.