Past Exam Questions

To express \(4x^2 - 12x + 13\) in the form \((2x + a)^2 + b\), we start by expanding \((2x + a)^2\):

\((2x + a)^2 = (2x)^2 + 2 \cdot 2x \cdot a + a^2 = 4x^2 + 4ax + a^2\).

We want this to match \(4x^2 - 12x + 13\). Comparing coefficients, we have:

\(4ax = -12x\) implies \(4a = -12\), so \(a = -3\).

Substitute \(a = -3\) into \(a^2\):

\(a^2 = (-3)^2 = 9\).

Now, compare the constant terms:

\(a^2 + b = 13\) implies \(9 + b = 13\), so \(b = 4\).

Thus, the expression is \((2x - 3)^2 + 4\).

To express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\), we complete the square:

1. Start with the expression: \(x^2 - 4x + 8\).

2. Take the coefficient of \(x\), which is \(-4\), halve it to get \(-2\), and square it to get \(4\).

3. Add and subtract \(4\) inside the expression:

\(x^2 - 4x + 4 - 4 + 8\)

4. Rewrite as a perfect square and simplify:

\((x-2)^2 + 4\)

To express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), follow these steps:

1. Factor out the coefficient of \(x^2\) from the quadratic and linear terms:

\(3(x^2 - 4x) + 7\)

2. Complete the square for the expression inside the parentheses:

\(x^2 - 4x = (x - 2)^2 - 4\)

3. Substitute back into the expression:

\(3((x - 2)^2 - 4) + 7\)

4. Distribute the 3 and simplify:

\(3(x - 2)^2 - 12 + 7\)

5. Combine the constants:

\(3(x - 2)^2 - 5\)

Thus, \(a = 3\), \(b = -2\), and \(c = -5\).

To express \(x^2 + 6x + 2\) in the form \((x + a)^2 + b\), we complete the square:

1. Start with the expression: \(x^2 + 6x + 2\).

2. Take half of the coefficient of \(x\), which is 6, giving 3, and square it: \(3^2 = 9\).

3. Add and subtract this square inside the expression: \(x^2 + 6x + 9 - 9 + 2\).

4. Rewrite as a perfect square: \((x + 3)^2 - 9 + 2\).

5. Simplify the constants: \((x + 3)^2 - 7\).

Thus, \(a = 3\) and \(b = -7\).

Given the function \(f(x) = x^2 + ax + b\) with roots \(x = 1\) and \(x = 9\), we can express it as:

\(f(x) = (x - 1)(x - 9)\)

Expanding this, we get:

\(f(x) = x^2 - 10x + 9\)

Thus, \(a = -10\) and \(b = 9\).

To find the vertex, use the vertex formula \(x = \frac{-b}{2a}\) or symmetry:

\(x = \frac{1 + 9}{2} = 5\)

Substitute \(x = 5\) into \(f(x)\):

\(f(5) = 5^2 - 10 \times 5 + 9 = -16\)

Thus, the vertex is \((5, -16)\).

To express \(2x^2 - 12x + 7\) in the form \(a(x+b)^2 + c\), we start by completing the square:

1. Factor out the coefficient of \(x^2\) from the first two terms:

\(2(x^2 - 6x) + 7\)

2. Complete the square inside the parentheses:

\(x^2 - 6x = (x-3)^2 - 9\)

3. Substitute back into the expression:

\(2((x-3)^2 - 9) + 7\)

4. Distribute the 2 and simplify:

\(2(x-3)^2 - 18 + 7\)

\(2(x-3)^2 - 11\)

Thus, \(a = 2\), \(b = -3\), and \(c = -11\).

To express \(2x^2 - 10x + 8\) in the form \(a(x + b)^2 + c\), we complete the square:

Start with the quadratic expression:

\(2x^2 - 10x + 8\)

Factor out the 2 from the first two terms:

\(2(x^2 - 5x) + 8\)

Complete the square inside the parentheses:

\(x^2 - 5x = (x - \frac{5}{2})^2 - (\frac{5}{2})^2\)

\(= (x - \frac{5}{2})^2 - \frac{25}{4}\)

Substitute back into the expression:

\(2((x - \frac{5}{2})^2 - \frac{25}{4}) + 8\)

\(= 2(x - \frac{5}{2})^2 - \frac{25}{2} + 8\)

\(= 2(x - \frac{5}{2})^2 - \frac{25}{2} + \frac{16}{2}\)

\(= 2(x - \frac{5}{2})^2 - \frac{9}{2}\)

Thus, \(a = 2\), \(b = -\frac{5}{2}\), \(c = -\frac{9}{2}\).

The minimum value of \(2x^2 - 10x + 8\) occurs at the vertex of the parabola, which is \(-\frac{9}{2}\).

To express \(4x^2 - 12x\) in the form \((2x + a)^2 + b\), we start by expanding \((2x + a)^2\):

\((2x + a)^2 = 4x^2 + 4ax + a^2\).

We want this to match \(4x^2 - 12x\), so we equate the coefficients:

\(4a = -12\) which gives \(a = -3\).

Substitute \(a = -3\) back into \((2x + a)^2\):

\((2x - 3)^2 = 4x^2 - 12x + 9\).

Now, compare \(4x^2 - 12x + 9\) with \(4x^2 - 12x\):

\(4x^2 - 12x = (2x - 3)^2 - 9\).

Thus, \(b = -9\).

To express \(2x^2 - 3x\) in the form \(a(x + b)^2 + c\), we complete the square:

Start with the equation:

\(y = 2x^2 - 3x\)

Factor out the 2 from the quadratic terms:

\(y = 2(x^2 - \frac{3}{2}x)\)

Complete the square inside the parentheses:

Take half of the coefficient of \(x\), square it, and add and subtract inside the bracket:

\(x^2 - \frac{3}{2}x = \left(x - \frac{3}{4}\right)^2 - \left(\frac{3}{4}\right)^2\)

Substitute back into the equation:

\(y = 2\left(\left(x - \frac{3}{4}\right)^2 - \frac{9}{16}\right)\)

Distribute the 2:

\(y = 2\left(x - \frac{3}{4}\right)^2 - \frac{9}{8}\)

Thus, the equation in the form \(a(x + b)^2 + c\) is:

\(y = 2\left(x - \frac{3}{4}\right)^2 - \frac{9}{8}\)

The vertex form \(a(x - h)^2 + k\) gives the vertex \((h, k)\).

So, the vertex is \(\left(\frac{3}{4}, -\frac{9}{8}\right)\).

To find the stationary point, we first differentiate \(f(x)\) with respect to \(x\):

\(f'(x) = \frac{d}{dx}[8 - (x - 2)^2] = -2(x - 2)\)

Set the derivative equal to zero to find the stationary point:

\(-2(x - 2) = 0\)

Solving for \(x\), we get:

\(x = 2\)

Substitute \(x = 2\) back into the original function to find the \(y\)-coordinate:

\(f(2) = 8 - (2 - 2)^2 = 8\)

Thus, the coordinates of the stationary point are \((2, 8)\).

To determine the nature of the stationary point, consider the second derivative:

\(f''(x) = \frac{d}{dx}[-2(x - 2)] = -2\)

Since \(f''(x) < 0\), the stationary point at \((2, 8)\) is a maximum.

To express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\), we complete the square:

Start with the expression:

\(x^2 - 4x + 5\)

Take half of the coefficient of \(x\), square it, and add and subtract it inside the expression:

\(x^2 - 4x + 4 - 4 + 5\)

Rewrite as a perfect square:

\((x - 2)^2 + 1\)

Thus, the expression in the form \((x + a)^2 + b\) is \((x - 2)^2 + 1\).

The minimum point occurs at \(x = 2\), giving the coordinates \((2, 1)\).

To express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), follow these steps:

1. Start with the expression: \(4x^2 - 24x + p\).

2. Factor out the 4 from the first two terms: \(4(x^2 - 6x) + p\).

3. Complete the square for \(x^2 - 6x\):

- Take half of the coefficient of \(x\), which is -6, to get -3.

- Square it to get 9.

4. Rewrite \(x^2 - 6x\) as \((x - 3)^2 - 9\).

5. Substitute back: \(4((x - 3)^2 - 9) + p\).

6. Expand and simplify: \(4(x - 3)^2 - 36 + p\).

7. Therefore, \(a = 4\), \(b = -3\), and \(c = p - 36\).

To express \(2x^2 - 4x + 1\) in the form \(a(x + b)^2 + c\), we complete the square:

Start with the quadratic expression:

\(2x^2 - 4x + 1\)

Factor out the 2 from the first two terms:

\(2(x^2 - 2x) + 1\)

Complete the square inside the parentheses:

\(x^2 - 2x = (x - 1)^2 - 1\)

Substitute back:

\(2((x - 1)^2 - 1) + 1\)

Distribute the 2:

\(2(x - 1)^2 - 2 + 1\)

Simplify:

\(2(x - 1)^2 - 1\)

Thus, \(a = 2\), \(b = -1\), \(c = -1\).

The minimum point \(A\) is at \((1, -1)\).

(a) To express \(y = 4x^2 + 20x + 6\) in the form \(y = a(x + b)^2 + c\), complete the square:

Factor out 4 from the quadratic terms: \(y = 4(x^2 + 5x) + 6\).

Complete the square for \(x^2 + 5x\):

\(x^2 + 5x = (x + \frac{5}{2})^2 - (\frac{5}{2})^2\).

Substitute back: \(y = 4((x + \frac{5}{2})^2 - \frac{25}{4}) + 6\).

Simplify: \(y = 4(x + \frac{5}{2})^2 - 25 + 6\).

\(y = 4(x + \frac{5}{2})^2 - 19\).

(b) Solve \(4(x + \frac{5}{2})^2 - 19 = 45\):

\(4(x + \frac{5}{2})^2 = 64\).

\((x + \frac{5}{2})^2 = 16\).

\(x + \frac{5}{2} = \pm 4\).

\(x = -\frac{5}{2} + 4 = \frac{3}{2}\) or \(x = -\frac{5}{2} - 4 = -\frac{13}{2}\).

(c) The stationary point is at \(x = -\frac{5}{2}\), giving \(y = -19\).

Stationary point: \((-\frac{5}{2}, -19)\).

(a) To express \(x^2 - 8x + 11\) in the form \((x + p)^2 + q\), complete the square:

Start with \(x^2 - 8x\).

Take half of the coefficient of \(x\), square it, and add and subtract it inside the expression:

\(x^2 - 8x = (x - 4)^2 - 16\).

Now add 11 to balance the equation:

\((x - 4)^2 - 16 + 11 = (x - 4)^2 - 5\).

Thus, \(p = -4\) and \(q = -5\).

(b) Solve \((x - 4)^2 - 5 = 1\).

First, add 5 to both sides:

\((x - 4)^2 = 6\).

Take the square root of both sides:

\(x - 4 = \pm \sqrt{6}\).

Finally, solve for \(x\):

\(x = 4 \pm \sqrt{6}\).

To express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\), we follow these steps:

1. Start with the expression: \(2x^2 - 8x + 14\).

2. Factor out the 2 from the first two terms: \(2(x^2 - 4x) + 14\).

3. Complete the square inside the parentheses:

- Take half of the coefficient of \(x\), which is \(-4\), giving \(-2\).

- Square it to get \(4\).

4. Add and subtract this square inside the parentheses:

\(2[(x^2 - 4x + 4) - 4] + 14\).

5. Simplify the expression:

\(2[(x-2)^2 - 4] + 14\).

6. Distribute the 2 and simplify:

\(2(x-2)^2 - 8 + 14\).

7. Combine the constants:

\(2(x-2)^2 + 6\).

8. Rewrite in the desired form:

\(2[(x-2)^2 + 3]\).

To express \(5y^2 - 30y + 50\) in the form \(5(y + a)^2 + b\), follow these steps:

1. Start with the expression: \(5y^2 - 30y + 50\).

2. Factor out 5 from the first two terms: \(5(y^2 - 6y) + 50\).

3. Complete the square inside the parentheses:

- Take half of the coefficient of \(y\), which is \(-6\), to get \(-3\).

- Square \(-3\) to get 9.

4. Add and subtract 9 inside the parentheses:

\(5(y^2 - 6y + 9 - 9) + 50\).

5. Rewrite the expression as:

\(5((y - 3)^2 - 9) + 50\).

6. Distribute the 5:

\(5(y - 3)^2 - 45 + 50\).

7. Simplify the constants:

\(5(y - 3)^2 + 5\).

Thus, \(a = -3\) and \(b = 5\).

To express \(16x^2 - 24x + 10\) in the form \((4x + a)^2 + b\), follow these steps:

1.

Start by expanding \((4x + a)^2\):

\((4x + a)^2 = 16x^2 + 8ax + a^2\) 2.

Compare coefficients with \(16x^2 - 24x + 10\):

- The coefficient of \(x\) gives: \(8a = -24\) - Solve for \(a\):

\(a = \frac{-24}{8} = -3\) 3.

Substitute \(a = -3\) back into \((4x + a)^2\):

\((4x - 3)^2 = 16x^2 - 24x + 9\) 4.

Find \(b\) by comparing the constant terms:

\(16x^2 - 24x + 10 = (4x - 3)^2 + b\) - From \(9 + b = 10\), solve for \(b\):

\(b = 1\) 5.

Thus, the expression is:

\((4x - 3)^2 + 1\)

To express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), we complete the square:

Start with the expression: \(x^2 + 6x + 5\).

1. Take half of the coefficient of \(x\), which is 6, giving 3, and square it: \(3^2 = 9\).

2. Add and subtract this square inside the expression:

\(x^2 + 6x + 9 - 9 + 5\).

3. Rewrite the expression as a perfect square and a constant:

\((x + 3)^2 - 4\).

Thus, \(a = 3\) and \(b = -4\).

Given the equation \(y = 2x^2 + kx + k - 1\) and \(k = 2\), substitute \(k\) to get:

\(y = 2x^2 + 2x + 2 - 1 = 2x^2 + 2x + 1\)

We need to express this in the form \(y = 2(x + a)^2 + b\).

Start by completing the square for \(2x^2 + 2x\):

\(2(x^2 + x) = 2\left((x + \frac{1}{2})^2 - \frac{1}{4}\right)\)

\(= 2(x + \frac{1}{2})^2 - \frac{1}{2}\)

Now, add the constant term:

\(y = 2(x + \frac{1}{2})^2 - \frac{1}{2} + 1\)

\(= 2(x + \frac{1}{2})^2 + \frac{1}{2}\)

Thus, \(a = \frac{1}{2}\) and \(b = \frac{1}{2}\).

The vertex form \(y = 2(x + a)^2 + b\) gives the vertex at \((-a, b)\).

Therefore, the vertex is \(\left(-\frac{1}{2}, \frac{1}{2}\right)\).