Question
determine the zeros of y = x^2 - 10x + 3
Answers
GPT-4o mini
To find the zeros of the quadratic function \( y = x^2 - 10x + 3 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -10 \), and \( c = 3 \).
1. Calculate the discriminant \( b^2 - 4ac \):
\[
b^2 - 4ac = (-10)^2 - 4(1)(3) = 100 - 12 = 88
\]
2. Now, substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[
x = \frac{-(-10) \pm \sqrt{88}}{2(1)} = \frac{10 \pm \sqrt{88}}{2}
\]
3. Simplify \( \sqrt{88} \):
\[
\sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22}
\]
4. Substitute this back into the formula:
\[
x = \frac{10 \pm 2\sqrt{22}}{2} = 5 \pm \sqrt{22}
\]
Thus, the two zeros of the equation \( y = x^2 - 10x + 3 \) are:
\[
x = 5 + \sqrt{22} \quad \text{and} \quad x = 5 - \sqrt{22}
\]
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -10 \), and \( c = 3 \).
1. Calculate the discriminant \( b^2 - 4ac \):
\[
b^2 - 4ac = (-10)^2 - 4(1)(3) = 100 - 12 = 88
\]
2. Now, substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[
x = \frac{-(-10) \pm \sqrt{88}}{2(1)} = \frac{10 \pm \sqrt{88}}{2}
\]
3. Simplify \( \sqrt{88} \):
\[
\sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22}
\]
4. Substitute this back into the formula:
\[
x = \frac{10 \pm 2\sqrt{22}}{2} = 5 \pm \sqrt{22}
\]
Thus, the two zeros of the equation \( y = x^2 - 10x + 3 \) are:
\[
x = 5 + \sqrt{22} \quad \text{and} \quad x = 5 - \sqrt{22}
\]