Answer :
To determine which of the given quadratic equations has the sum of its roots equal to 4, we can use a property of quadratic equations. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the sum of the roots (\( \alpha + \beta \)) is given by \( -\frac{b}{a} \).
Let's analyze each equation one-by-one to find the sum of the roots:
(a) \( 2x^2 - 4x + 8 = 0 \)
For this equation:
- \( a = 2 \)
- \( b = -4 \)
- The sum of the roots is \( \frac{-(-4)}{2} = \frac{4}{2} = 2.0 \)
(b) \( -x^2 + 4x + 4 = 0 \)
For this equation:
- \( a = -1 \)
- \( b = 4 \)
- The sum of the roots is \( \frac{-(4)}{-1} = \frac{-4}{-1} = 4.0 \)
(c) \( -\sqrt{2}x^2 - \frac{4}{\sqrt{2}}x + 1 = 0 \)
For this equation:
- \( a = -\sqrt{2} \)
- \( b = -\frac{4}{\sqrt{2}} \)
- The sum of the roots is \( \frac{-(-\frac{4}{\sqrt{2}})}{-\sqrt{2}} = \frac{\frac{4}{\sqrt{2}}}{-\sqrt{2}} = -2.0 \)
(d) \( 4x^2 - 4x + 4 = 0 \)
For this equation:
- \( a = 4 \)
- \( b = -4 \)
- The sum of the roots is \( \frac{-(-4)}{4} = \frac{4}{4} = 1.0 \)
After calculating the sum of the roots for each equation, we see that the quadratic equation in option (b), which is \( -x^2 + 4x + 4 = 0 \), has a sum of its roots equal to 4. Therefore, the correct answer is:
(b) [tex]\( -x^2 + 4x + 4 = 0 \)[/tex]
Let's analyze each equation one-by-one to find the sum of the roots:
(a) \( 2x^2 - 4x + 8 = 0 \)
For this equation:
- \( a = 2 \)
- \( b = -4 \)
- The sum of the roots is \( \frac{-(-4)}{2} = \frac{4}{2} = 2.0 \)
(b) \( -x^2 + 4x + 4 = 0 \)
For this equation:
- \( a = -1 \)
- \( b = 4 \)
- The sum of the roots is \( \frac{-(4)}{-1} = \frac{-4}{-1} = 4.0 \)
(c) \( -\sqrt{2}x^2 - \frac{4}{\sqrt{2}}x + 1 = 0 \)
For this equation:
- \( a = -\sqrt{2} \)
- \( b = -\frac{4}{\sqrt{2}} \)
- The sum of the roots is \( \frac{-(-\frac{4}{\sqrt{2}})}{-\sqrt{2}} = \frac{\frac{4}{\sqrt{2}}}{-\sqrt{2}} = -2.0 \)
(d) \( 4x^2 - 4x + 4 = 0 \)
For this equation:
- \( a = 4 \)
- \( b = -4 \)
- The sum of the roots is \( \frac{-(-4)}{4} = \frac{4}{4} = 1.0 \)
After calculating the sum of the roots for each equation, we see that the quadratic equation in option (b), which is \( -x^2 + 4x + 4 = 0 \), has a sum of its roots equal to 4. Therefore, the correct answer is:
(b) [tex]\( -x^2 + 4x + 4 = 0 \)[/tex]
