Since the y coordinate of the vertex is greater than zero, the quadratic equation has no real roots
To show that the quadratic equation -2x + 8x - 13 = 0 has no real roots, we desire to write it in the form a(x+p)² + q.
To write it in this form, we complete the square
So, -2x² + 8x - 13 = 0
Dividing through by -2, we have
-2x²/-2 + 8x/-2 -13/-2 = 0/-2
x² - 4x + 13/2 = 0
Adding the square of half the coefficient of x to both sides, we have
x² - 4x + 13/2 + (-4/2)² = 0 + (-4/2)²
x² - 4x + 13/2 + (-2)² = 0 + (-2)²
x² - 4x + 13/2 + 2² = 0 + 2²
x² - 4x + 2² + 13/2 + = 0 + 2²
(x - 2)² + 13/2 = 4
(x - 2)² + 13/2 - 4 = 0
(x - 2)² + (13 - 8)/2 = 0
(x - 2)² + 5/2 = 0
2(x - 2)² + 5 = 0
Since the equation is now written in vertex form with vertex at (2, 5), we see that the y coordinate of the vertex is greater than zero. Thus, the graph does not cross the x - axis and thus has no real roots
Since the y coordinate of the vertex is greater than zero, the quadratic equation has no real roots
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