To solve the equation [tex]\( y^2 = 10y + 25 \)[/tex] by factoring, let's first rewrite the equation in standard quadratic form, which is [tex]\( ax^2 + bx + c = 0 \)[/tex].
1. Start with the equation:
[tex]\[ y^2 = 10y + 25 \][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[ y^2 - 10y - 25 = 0 \][/tex]
Next, we will try to solve this equation by factoring. However, factoring is often easier when dealing with simpler numbers. If factoring doesn’t readily present a solution, the quadratic formula can be used. The quadratic formula is given by:
[tex]\[ y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
For the equation [tex]\( y^2 - 10y - 25 = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -10 \)[/tex]
- [tex]\( c = -25 \)[/tex]
3. Substitute these values into the quadratic formula:
[tex]\[ y = \frac{{-(-10) \pm \sqrt{{(-10)^2 - 4 \cdot 1 \cdot (-25)}}}}{2 \cdot 1} \][/tex]
4. Simplify inside the square root:
[tex]\[ y = \frac{{10 \pm \sqrt{{100 + 100}}}}{2} \][/tex]
[tex]\[ y = \frac{{10 \pm \sqrt{{200}}}}{2} \][/tex]
5. Simplify the square root and the fraction:
[tex]\[ y = \frac{{10 \pm 10\sqrt{2}}}{2} \][/tex]
[tex]\[ y = 5 \pm 5\sqrt{2} \][/tex]
Thus, the solutions to the equation [tex]\( y^2 = 10y + 25 \)[/tex] are:
[tex]\[ y = 5 - 5\sqrt{2} \][/tex]
[tex]\[ y = 5 + 5\sqrt{2} \][/tex]
So, the solutions are:
[tex]\[ y = 5 - 5\sqrt{2} \][/tex]
[tex]\[ y = 5 + 5\sqrt{2} \][/tex]
