Factorising Quadratic Equations Sheet 8 - Practice problems for factoring quadratic equations.

To solve the problem of factorizing quadratic equations, we need to express each quadratic equation in the form \((x - p)(x - q) = 0\), where \(p\) and \(q\) are the roots of the equation. The general steps are:

1. Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \(ax^2 + bx + c = 0\).
2. Find two numbers whose product is \(ac\) and whose sum is \(b\).
3. Rewrite the middle term using these two numbers.
4. Factor by grouping.
5. Solve for the roots.

Let's go through each equation step by step.

Problem 1: \(x^2 - 10x + 24 = 0\)

1. Identify \(a = 1\), \(b = -10\), and \(c = 24\).
2. Find two numbers whose product is \(1 \cdot 24 = 24\) and whose sum is \(-10\). These numbers are \(-6\) and \(-4\).
3. Rewrite the equation: \(x^2 - 6x - 4x + 24 = 0\).
4. Factor by grouping: \(x(x - 6) - 4(x - 6) = 0\).
5. Factor out the common term: \((x - 6)(x - 4) = 0\).
6. Solve for \(x\): \(x = 6\) or \(x = 4\).

So, the factorized form is \((x - 6)(x - 4) = 0\) and the solutions are \(x = 6\) or \(x = 4\).

Problem 2: \(y^2 - 15y + 14 = 0\)

1. Identify \(a = 1\), \(b = -15\), and \(c = 14\).
2. Find two numbers whose product is \(1 \cdot 14 = 14\) and whose sum is \(-15\). These numbers are \(-1\) and \(-14\).
3. Rewrite the equation: \(y^2 - y - 14y + 14 = 0\).
4. Factor by grouping: \(y(y - 1) - 14(y - 1) = 0\).
5. Factor out the common term: \((y - 1)(y - 14) = 0\).
6. Solve for \(y\): \(y = 1\) or \(y = 14\).

So, the factorized form is \((y - 1)(y - 14) = 0\) and the solutions are \(y = 1\) or \(y = 14\).

Problem 3: \(z^2 - 8z - 20 = 0\)

1. Identify \(a = 1\), \(b = -8\), and \(c = -20\).
2. Find two numbers whose product is \(1 \cdot (-20) = -20\) and whose sum is \(-8\). These numbers are \(-10\) and \(2\).
3. Rewrite the equation: \(z^2 - 10z + 2z - 20 = 0\).
4. Factor by grouping: \(z(z - 10) + 2(z - 10) = 0\).
5. Factor out the common term: \((z - 10)(z + 2) = 0\).
6. Solve for \(z\): \(z = 10\) or \(z = -2\).

So, the factorized form is \((z - 10)(z + 2) = 0\) and the solutions are \(z = 10\) or \(z = -2\).

Problem 4: \(a^2 - 7a - 30 = 0\)

1. Identify \(a = 1\), \(b = -7\), and \(c = -30\).
2. Find two numbers whose product is \(1 \cdot (-30) = -30\) and whose sum is \(-7\). These numbers are \(-10\) and \(3\).
3. Rewrite the equation: \(a^2 - 10a + 3a - 30 = 0\).
4. Factor by grouping: \(a(a - 10) + 3(a - 10) = 0\).
5. Factor out the common term: \((a - 10)(a + 3) = 0\).
6. Solve for \(a\): \(a = 10\) or \(a = -3\).

So, the factorized form is \((a - 10)(a + 3) = 0\) and the solutions are \(a = 10\) or \(a = -3\).

Problem 5: \(b^2 - 16b + 28 = 0\)

1. Identify \(a = 1\), \(b = -16\), and \(c = 28\).
2. Find two numbers whose product is \(1 \cdot 28 = 28\) and whose sum is \(-16\). These numbers are \(-14\) and \(-2\).
3. Rewrite the equation: \(b^2 - 14b - 2b + 28 = 0\).
4. Factor by grouping: \(b(b - 14) - 2(b - 14) = 0\).
5. Factor out the common term: \((b - 14)(b - 2) = 0\).
6. Solve for \(b\): \(b = 14\) or \(b = 2\).

So, the factorized form is \((b - 14)(b - 2) = 0\) and the solutions are \(b = 14\) or \(b = 2\).

Problem 6: \(c^2 + 15c - 34 = 0\)

1. Identify \(a = 1\), \(b = 15\), and \(c = -34\).
2. Find two numbers whose product is \(1 \cdot (-34) = -34\) and whose sum is \(15\). These numbers are \(17\) and \(-2\).
3. Rewrite the equation: \(c^2 + 17c - 2c - 34 = 0\).
4. Factor by grouping: \(c(c + 17) - 2(c + 17) = 0\).
5. Factor out the common term: \((c + 17)(c - 2) = 0\).
6. Solve for \(c\): \(c = -17\) or \(c = 2\).

So, the factorized form is \((c + 17)(c - 2) = 0\) and the solutions are \(c = -17\) or \(c = 2\).

Problem 7: \(d^2 - 9d - 36 = 0\)

1. Identify \(a = 1\), \(b = -9\), and \(c = -36\).
2. Find two numbers whose product is \(1 \cdot (-36) = -36\) and whose sum is \(-9\). These numbers are \(-12\) and \(3\).
3. Rewrite the equation: \(d^2 - 12d + 3d - 36 = 0\).
4. Factor by grouping: \(d(d - 12) + 3(d - 12) = 0\).
5. Factor out the common term: \((d - 12)(d + 3) = 0\).
6. Solve for \(d\): \(d = 12\) or \(d = -3\).

So, the factorized form is \((d - 12)(d + 3) = 0\) and the solutions are \(d = 12\) or \(d = -3\).

Problem 8: \(e^2 - 10e + 25 = 0\)

1. Identify \(a = 1\), \(b = -10\), and \(c = 25\).
2. Find two numbers whose product is \(1 \cdot 25 = 25\) and whose sum is \(-10\). These numbers are \(-5\) and \(-5\).
3. Rewrite the equation: \(e^2 - 5e - 5e + 25 = 0\).
4. Factor by grouping: \(e(e - 5) - 5(e - 5) = 0\).
5. Factor out the common term: \((e - 5)(e - 5) = 0\).
6. Solve for \(e\): \(e = 5\) or \(e = 5\).

So, the factorized form is \((e - 5)^2 = 0\) and the solution is \(e = 5\).

Problem 9: \(f^2 - 64 = 0\)

1. This is a difference of squares: \(f^2 - 64 = (f - 8)(f + 8) = 0\).
2. Solve for \(f\): \(f = 8\) or \(f = -8\).

So, the factorized form is \((f - 8)(f + 8) = 0\) and the solutions are \(f = 8\) or \(f = -8\).

Problem 10: \(g^2 - 11g - 42 = 0\)

1. Identify \(a = 1\), \(b = -11\), and \(c = -42\).
2. Find two numbers whose product is \(1 \cdot (-42) = -42\) and whose sum is \(-11\). These numbers are \(-14\) and \(3\).
3. Rewrite the equation: \(g^2 - 14g + 3g - 42 = 0\).
4. Factor by grouping: \(g(g - 14) + 3(g - 14) = 0\).
5. Factor out the common term: \((g - 14)(g + 3) = 0\).
6. Solve for \(g\): \(g = 14\) or \(g = -3\).

So, the factorized form is \((g - 14)(g + 3) = 0\) and the solutions are \(g = 14\) or \(g = -3\).

Problem 11: \(h^2 + 22h + 40 = 0\)

1. Identify \(a = 1\), \(b = 22\), and \(c = 40\).
2. Find two numbers whose product is \(1 \cdot 40 = 40\) and whose sum is \(22\). These numbers are \(20\) and \(2\).
3. Rewrite the equation: \(h^2 + 20h + 2h + 40 = 0\).
4. Factor by grouping: \(h(h + 20) + 2(h + 20) = 0\).
5. Factor out the common term: \((h + 20)(h + 2) = 0\).
6. Solve for \(h\): \(h = -20\) or \(h = -2\).

So, the factorized form is \((h + 20)(h + 2) = 0\) and the solutions are \(h = -20\) or \(h = -2\).

Problem 12: \(i^2 - 14i - 32 = 0\)

1. Identify \(a = 1\), \(b = -14\), and \(c = -32\).
2. Find two numbers whose product is \(1 \cdot (-32) = -32\) and whose sum is \(-14\). These numbers are \(-16\) and \(2\).
3. Rewrite the equation: \(i^2 - 16i + 2i - 32 = 0\).
4. Factor by grouping: \(i(i - 16) + 2(i - 16) = 0\).
5. Factor out the common term: \((i - 16)(i + 2) = 0\).
6. Solve for \(i\): \(i = 16\) or \(i = -2\).

So, the factorized form is \((i - 16)(i + 2) = 0\) and the solutions are \(i = 16\) or \(i = -2\).

Problem 13: \(j^2 - 13j + 42 = 0\)

1. Identify \(a = 1\), \(b = -13\), and \(c = 42\).
2. Find two numbers whose product is \(1 \cdot 42 = 42\) and whose sum is \(-13\). These numbers are \(-7\) and \(-6\).
3. Rewrite the equation: \(j^2 - 7j - 6j + 42 = 0\).
4. Factor by grouping: \(j(j - 7) - 6(j - 7) = 0\).
5. Factor out the common term: \((j - 7)(j - 6) = 0\).
6. Solve for \(j\): \(j = 7\) or \(j = 6\).

So, the factorized form is \((j - 7)(j - 6) = 0\) and the solutions are \(j = 7\) or \(j = 6\).

Problem 14: \(k^2 + 3k - 54 = 0\)

1. Identify \(a = 1\), \(b = 3\), and \(c = -54\).
2. Find two numbers whose product is \(1 \cdot (-54) = -54\) and whose sum is \(3\). These numbers are \(9\) and \(-6\).
3. Rewrite the equation: \(k^2 + 9k - 6k - 54 = 0\).
4. Factor by grouping: \(k(k + 9) - 6(k + 9) = 0\).
5. Factor out the common term: \((k + 9)(k - 6) = 0\).
6. Solve for \(k\): \(k = -9\) or \(k = 6\).

So, the factorized form is \((k + 9)(k - 6) = 0\) and the solutions are \(k = -9\) or \(k = 6\).

Problem 15: \(m^2 + 12m - 45 = 0\)

1. Identify \(a = 1\), \(b = 12\), and \(c = -45\).
2. Find two numbers whose product is \(1 \cdot (-45) = -45\) and whose sum is \(12\). These numbers are \(15\) and \(-3\).
3. Rewrite the equation: \(m^2 + 15m - 3m - 45 = 0\).
4. Factor by grouping: \(m(m + 15) - 3(m + 15) = 0\).
5. Factor out the common term: \((m + 15)(m - 3) = 0\).
6. Solve for \(m\): \(m = -15\) or \(m = 3\).

So, the factorized form is \((m + 15)(m - 3) = 0\) and the solutions are \(m = -15\) or \(m = 3\).

Problem 16: \(n^2 - 13n + 40 = 0\)

1. Identify \(a = 1\), \(b = -13\), and \(c = 40\).
2. Find two numbers whose product is \(1 \cdot 40 = 40\) and whose sum is \(-13\). These numbers are \(-8\) and \(-5\).
3. Rewrite the equation: \(n^2 - 8n - 5n + 40 = 0\).
4. Factor by grouping: \(n(n - 8) - 5(n - 8) = 0\).
5. Factor out the common term: \((n - 8)(n - 5) = 0\).
6. Solve for \(n\): \(n = 8\) or \(n = 5\).

So, the factorized form is \((n - 8)(n - 5) = 0\) and the solutions are \(n = 8\) or \(n = 5\).

Final Answer

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
1 & x^2 - 10x + 24 = 0 & (x - 6)(x - 4) = 0 & x = 6 \text{ or } x = 4 \\
\hline
2 & y^2 - 15y + 14 = 0 & (y - 1)(y - 14) = 0 & y = 1 \text{ or } y = 14 \\
\hline
3 & z^2 - 8z - 20 = 0 & (z - 10)(z + 2) = 0 & z = 10 \text{ or } z = -2 \\
\hline
4 & a^2 - 7a - 30 = 0 & (a - 10)(a + 3) = 0 & a = 10 \text{ or } a = -3 \\
\hline
5 & b^2 - 16b + 28 = 0 & (b - 14)(b - 2) = 0 & b = 14 \text{ or } b = 2 \\
\hline
6 & c^2 + 15c - 34 = 0 & (c + 17)(c - 2) = 0 & c = -17 \text{ or } c = 2 \\
\hline
7 & d^2 - 9d - 36 = 0 & (d - 12)(d + 3) = 0 & d = 12 \text{ or } d = -3 \\
\hline
8 & e^2 - 10e + 25 = 0 & (e - 5)^2 = 0 & e = 5 \\
\hline
9 & f^2 - 64 = 0 & (f - 8)(f + 8) = 0 & f = 8 \text{ or } f = -8 \\
\hline
10 & g^2 - 11g - 42 = 0 & (g - 14)(g + 3) = 0 & g = 14 \text{ or } g = -3 \\
\hline
11 & h^2 + 22h + 40 = 0 & (h + 20)(h + 2) = 0 & h = -20 \text{ or } h = -2 \\
\hline
12 & i^2 - 14i - 32 = 0 & (i - 16)(i + 2) = 0 & i = 16 \text{ or } i = -2 \\
\hline
13 & j^2 - 13j + 42 = 0 & (j - 7)(j - 6) = 0 & j = 7 \text{ or } j = 6 \\
\hline
14 & k^2 + 3k - 54 = 0 & (k + 9)(k - 6) = 0 & k = -9 \text{ or } k = 6 \\
\hline
15 & m^2 + 12m - 45 = 0 & (m + 15)(m - 3) = 0 & m = -15 \text{ or } m = 3 \\
\hline
16 & n^2 - 13n + 40 = 0 & (n - 8)(n - 5) = 0 & n = 8 \text{ or } n = 5 \\
\hline
\end{array}
}
\]