1.1

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

Choose the right answer.

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

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 \(p \pm q\sqrt{11}\).

Choose the right answer.

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

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 \(y(-1.25) = 2(-1.25 + 1.25)^2 - 3.125 = -3.125\).