Question

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$b^2 - 4ac$ is called the discriminant
$b^2 - 4ac$ is zero $\
ightarrow$ 1 real solution
$b^2 - 4ac$ is positive $\
ightarrow$ 2 real solutions
$x^2 -10x +25 = 0$
$= 10\pm\sqrt{-103 -1(1)}$
$x^2 -9 = 0$
$x^2 +6x +2 = 0$

Explanation:

Step1: Identify discriminant rule

For $ax^2+bx+c=0$, discriminant $D=b^2-4ac$

• $D=0$: 1 real solution
• $D>0$: 2 real solutions

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For $x^2-10x+25=0$

Step2: Assign a,b,c values

$a=1,\ b=-10,\ c=25$

Step3: Calculate discriminant

$D=(-10)^2-4(1)(25)=100-100=0$

Step4: Determine solution count

Since $D=0$, 1 real solution

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For $x^2-9=0$

Step5: Assign a,b,c values

$a=1,\ b=0,\ c=-9$

Step6: Calculate discriminant

$D=(0)^2-4(1)(-9)=0+36=36$

Step7: Determine solution count

Since $D>0$, 2 real solutions

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For $x^2+6x+2=0$

Step8: Assign a,b,c values

$a=1,\ b=6,\ c=2$

Step9: Calculate discriminant

$D=(6)^2-4(1)(2)=36-8=28$

Step10: Determine solution count

Since $D>0$, 2 real solutions

Answer:

• $x^2-10x+25=0$: 1 real solution
• $x^2-9=0$: 2 real solutions
• $x^2+6x+2=0$: 2 real solutions