Answer :

To solve the equation [tex]\( y^2 = 28y - 187 \)[/tex], we need to rearrange it into a standard quadratic form and then solve for [tex]\( y \)[/tex].

1. Start with the given equation:

[tex]\[
y^2 = 28y - 187
\][/tex]

2. Rearrange the equation:

Move all the terms to one side of the equation to set it to zero:

[tex]\[
y^2 - 28y + 187 = 0
\][/tex]

This is now a quadratic equation in the form of [tex]\( ay^2 + by + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -28 \)[/tex], and [tex]\( c = 187 \)[/tex].

3. Solve the quadratic equation:

We use the quadratic formula, which is:

[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

Plugging in the values [tex]\( a = 1 \)[/tex], [tex]\( b = -28 \)[/tex], and [tex]\( c = 187 \)[/tex]:

[tex]\[
y = \frac{-(-28) \pm \sqrt{(-28)^2 - 4 \cdot 1 \cdot 187}}{2 \cdot 1}
\][/tex]

4. Calculate the discriminant:

[tex]\[
(-28)^2 - 4 \cdot 1 \cdot 187 = 784 - 748 = 36
\][/tex]

5. Substitute the discriminant back into the formula:

[tex]\[
y = \frac{28 \pm \sqrt{36}}{2}
\][/tex]

6. Calculate the square root and simplify:

[tex]\[
\sqrt{36} = 6
\][/tex]

So we have:

[tex]\[
y = \frac{28 \pm 6}{2}
\][/tex]

7. Find the two possible values for [tex]\( y \)[/tex]:

- For [tex]\( y = \frac{28 + 6}{2} \)[/tex]:

[tex]\[
y = \frac{34}{2} = 17
\][/tex]

- For [tex]\( y = \frac{28 - 6}{2} \)[/tex]:

[tex]\[
y = \frac{22}{2} = 11
\][/tex]

So, the solutions to the equation [tex]\( y^2 = 28y - 187 \)[/tex] are [tex]\( y = 11 \)[/tex] and [tex]\( y = 17 \)[/tex].

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