Solve for \( y \). Try factoring first. If factoring is not possible or is difficult, use the quadratic formula.

[tex]\[ y^2 - 10y = -23 \][/tex]



Answer :

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]