Question

the following equations. write any non - integer answers in exact and simplified form.
combined are combined, terms that cannot be combined are not combined
off decimals in any of my answers.
s or decimals inside any of the square roots in any of my answers.
roots in the denominators in any of my answers.
roots that can be simplified further in any of my answers.
y answers are written in the form a + bi, a - bi, or a ± bi.
10.) $x^2 - 25 = 0$
$x = \pm 5$
12.) $x^2 + 4x + 5 = 0$
$x = -2 \pm i$
14.) $x^2 + 6x - 1 = 0$

Explanation:

Step1: Solve \(x^2 - 25 = 0\)

We can rewrite the equation as a difference of squares: \(x^2 - 5^2 = 0\). Using the formula \(a^2 - b^2=(a + b)(a - b)\), we get \((x + 5)(x - 5)=0\).
Setting each factor equal to zero gives \(x+5 = 0\) or \(x - 5 = 0\).
Solving \(x+5 = 0\) gives \(x=-5\), and solving \(x - 5 = 0\) gives \(x = 5\).

Step2: Solve \(x^2+4x + 5 = 0\)

We use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) for a quadratic equation \(ax^2+bx + c = 0\). Here, \(a = 1\), \(b = 4\), and \(c = 5\).
First, calculate the discriminant \(\Delta=b^2-4ac=(4)^2-4\times1\times5=16 - 20=-4\).
Then, \(x=\frac{-4\pm\sqrt{-4}}{2\times1}=\frac{-4\pm2i}{2}=-2\pm i\).

Step3: Solve \(x^2+6x - 1 = 0\)

Using the quadratic formula with \(a = 1\), \(b = 6\), and \(c=-1\).
The discriminant \(\Delta=b^2-4ac=(6)^2-4\times1\times(-1)=36 + 4 = 40\).
Then, \(x=\frac{-6\pm\sqrt{40}}{2\times1}=\frac{-6\pm2\sqrt{10}}{2}=-3\pm\sqrt{10}\).

Answer:

• For \(x^2 - 25 = 0\): \(x = 5\) or \(x=-5\)
• For \(x^2+4x + 5 = 0\): \(x=-2 + i\) or \(x=-2 - i\)
• For \(x^2+6x - 1 = 0\): \(x=-3+\sqrt{10}\) or \(x=-3-\sqrt{10}\)