Sure! Let's go through each question step-by-step.
1.) Solve \(x^2 + 20x + 99\)
To solve this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 20\), and \(c = 99\).
Calculating the discriminant:
\[ b^2 - 4ac = 20^2 - 4(1)(99) = 400 - 396 = 4 \]
Now, substituting values into the formula:
\[ x = \frac{-20 \pm \sqrt{4}}{2(1)} = \frac{-20 \pm 2}{2} \]
The solutions are:
\[ x = \frac{-20 + 2}{2} = \frac{-18}{2} = -9 \] \[ x = \frac{-20 - 2}{2} = \frac{-22}{2} = -11 \]
Thus, the solutions are \(x = -9\) and \(x = -11\).
2.) Factor the expression \(x^2 + 15x + 44\)
To factor the trinomial, we look for two numbers that multiply to \(44\) and add to \(15\). These numbers are \(11\) and \(4\).
Thus, the factored form is:
\[ (x + 11)(x + 4) \]
3.) Factor the expression \(n^2 - 14n + 40\)
We need two numbers that multiply to \(40\) and add to \(-14\). These are \(-10\) and \(-4\).
Thus, the factored form is:
\[ (n - 10)(n - 4) \]
4.) Factor the expression \(n^2 - 30n + 56\)
We need two numbers that multiply to \(56\) and add to \(-30\). These are \(-28\) and \(-2\).
Thus, the factored form is:
\[ (n - 28)(n - 2) \]
5.) Factor the trinomial completely: \(x^2 + x - 20\)
We look for two numbers that multiply to \(-20\) and add to \(1\). These numbers are \(5\) and \(-4\).
The factored form is:
\[ (x + 5)(x - 4) \]
6.) Factor the trinomial completely: \(x^2 + 2x - 35\)
We need two numbers that multiply to \(-35\) and add to \(2\). These numbers are \(7\) and \(-5\).
The factored form is:
\[ (x + 7)(x - 5) \]
7.) Factor the trinomial: \(c^2 + 7cz + 12z^2\)
To factor this quadratic, we look for two numbers that multiply to \(12\) and add to \(7\). These are \(3\) and \(4\).
Thus, we can factor by grouping:
\[ c^2 + 7cz + 12z^2 = (c + 3z)(c + 4z) \]
8.) Factor the trinomial: \(s^2 - sf - 72f^2\)
We need two numbers that multiply to \(-72\) and add to \(-1\). These numbers are \(8\) and \(-9\).
Thus, we can factor:
\[ s^2 - sf - 72f^2 = (s - 9f)(s + 8f) \]
9.) Factor the expression \(x^2 + 2xn + n^2\)
This expression is a perfect square trinomial. It can be factored as:
\[ x^2 + 2xn + n^2 = (x + n)^2 \]
10.) The area of a rectangular rug is given by the trinomial \(r^2 - 4r - 77\). What are the possible dimensions of the rug?
We need to factor \(r^2 - 4r - 77\). We look for two numbers that multiply to \(-77\) and add to \(-4\). These numbers are \(-11\) and \(7\).
Thus, we have:
\[ r^2 - 4r - 77 = (r - 11)(r + 7) \]
The possible dimensions of the rug are \(r - 11\) and \(r + 7\). Since dimensions cannot be negative, we are specifically interested in when \(r - 11 > 0\), or \(r > 11\).
So, possible dimensions are:
\[ (r - 11) \text{ and } (r + 7) \text{ where } r > 11. \]
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