-5x2 + 20x - 13 describes a parabola, and you can find it's vertex in various ways. To find it using calculus, you can simply take it's derivative and solve for 0:
f(x) = -5x2 + 20x - 13
f'(x) = -10x + 20
Let f'(x) = 0, then:
0 = -10x + 20
x = 2
Then simply plug the x value into the original equation:
f(x) = -5 * 22 + 20 * 2 - 13
= -20 + 40 - 13
= 7
So the vertex of this parabola is at the point (2, 7).
To solve it using strictly algebra, you can do it by expressing it as a value of y, and then rearranging accordingly:
y = -5x2 + 20x - 13
y = -5(x2 - 4x) - 13
y = -5(x2 - 4x + 4 - 4) - 13
y = -5(x2 - 4x + 4) + 20 - 13
y = -5(x - 2)2 + 7
From which we can see that the vertex happens at the point (2, 7)
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5x2 + 9x + 4 = (x + 1)(5x + 4).
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Assuming that the 2 in "5x2" is a power (5x2), then no, this is not a linear equation. It is a parabolic equation.
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