Let's start with the quadratic equation \( y^2 - 10y = -23 \). First, we will rearrange this equation to standard quadratic form, which is \( ay^2 + by + c = 0 \).
[tex]\[
y^2 - 10y + 23 = 0
\][/tex]
### Step 1: Attempt to Factor the Quadratic Equation
We need to find two numbers that multiply to \( 23 \) (the constant term) and add up to \( -10 \) (the coefficient of \( y \)).
Let's examine the factors of 23:
- The number 23 is a prime number, so it has only two factors: \( 1 \) and \( 23 \).
There are no pairs of numbers that satisfy both conditions (multiplying to 23 and adding to -10). Therefore, factoring does not seem feasible in this case.
### Step 2: Use the Quadratic Formula
Since factoring is not possible, we will use the quadratic formula given by:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this equation, \( a = 1 \), \( b = -10 \), and \( c = 23 \).
### Step 3: Calculate the Discriminant
The discriminant \(\Delta\) is an important part of the quadratic formula and is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values:
[tex]\[
\Delta = (-10)^2 - 4 \cdot 1 \cdot 23 = 100 - 92 = 8
\][/tex]
Since the discriminant is positive (\(\Delta = 8\)), we have two distinct real solutions.
### Step 4: Find the Solutions
Using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute \( b = -10 \), \( a = 1 \), and \(\Delta = 8\):
[tex]\[
y = \frac{-(-10) \pm \sqrt{8}}{2 \cdot 1} = \frac{10 \pm 2\sqrt{2}}{2}
\][/tex]
Simplify the expression:
[tex]\[
y = \frac{10}{2} \pm \frac{2\sqrt{2}}{2} = 5 \pm \sqrt{2}
\][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[
y_1 = 5 + \sqrt{2} \approx 6.414213562373095
\][/tex]
[tex]\[
y_2 = 5 - \sqrt{2} \approx 3.585786437626905
\][/tex]
### Summary
The quadratic equation \( y^2 - 10y + 23 = 0 \) has two real solutions:
[tex]\[
y_1 \approx 6.414213562373095
\][/tex]
[tex]\[
y_2 \approx 3.585786437626905
\][/tex]
And the discriminant is:
[tex]\[
\Delta = 8
\][/tex]
