How do you find the discriminant and how many solutions does $- 5 x^{2} + 7 x - 2 = 0$ $-5x^2+7x-2=0$ have?

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How do you find the discriminant and how many solutions does $- 5 x^{2} + 7 x - 2 = 0$ $-5x^2+7x-2=0$ have?

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Answer 1 · 2015-04-30T19:15:06

Discriminant $D = d^{2} = b^{2} - 4 a c = 49 - 40 = 9$ $D = d^2 = b^2 - 4ac = 49 - 40 = 9$ --> d = 3 and d = -3.
There are 2 solutions, called real roots of the quadratic equations.

$x 1 = - \frac{b}{2 a} + \frac{d}{2 a} = \frac{7}{10} - \frac{3}{10} = \frac{4}{10} = \frac{2}{5}$ $x1 = -b/(2a) + d/(2a) = 7/10 - 3/10 = 4/10 = 2/5$

$x 2 = - \frac{b}{2} a - \frac{d}{2} a = \frac{7}{10} + \frac{3}{10} = \frac{10}{10} = 1$ $x2 = -b/2a - d/2a = 7/10 + 3/10 = 10/10 = 1$

Note . We can use shortcut to quickly solve this equation.

Since a + b + c = 0, then one real root is 1 and the other is $(\frac{c}{a} = \frac{- 2}{- 5} = \frac{2}{5})$ $(c/a = (-2)/-5 = 2/5)$ .
When a - b + c = 0, then one real root is (-1) and the other is $(- \frac{c}{a})$ $(-c/a)$ .

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How do you find the discriminant and how many solutions does $- 5 x^{2} + 7 x - 2 = 0$ $-5x^2+7x-2=0$ have?

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