Sirs-please-give-me-the-general-solutions-to-a-quadratic-ineqality-1-ax-2-bx-c-gt-0-2-ax-2-bx-c-0-3-ax-2-bx-c-lt-0

Question Number 156652 by Tawa11 last updated on 13/Oct/21

Sirs, please give me the general solutions to a quadratic ineqality .

(1) {ax}^{2} + bx + c > 0

(2) {ax}^{2} + bx + c ⩾ 0

(3) {ax}^{2} + bx + c < 0

(4) {ax}^{2} + bx + c ⩽ 0

Answered by MJS_new last updated on 14/Oct/21

f (x) = {a x}^{2} + b x + c

1 . solve the equation

{a x}^{2} + b x + c = 0

\Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

2 . how many real solutions ?

2.1 . b^{2} - 4 a c > 0 \Rightarrow 2 real solutions x_{1} < x_{2}

2.2 . b^{2} - 4 a c = 0 \Rightarrow 1 real solution x_{1}

2.3 . b^{2} - 4 a c < 0 \Rightarrow no real solution

3 . check a

3.1 . a < 0 \Rightarrow “ {hanging}^{″} parabola

3.2 . a = 0 \Rightarrow straight line

3.3 . a > 0 \Rightarrow “ {standing}^{″} parabola

\Rightarrow

all together

a < 0 \land b^{2} - 4 a c > 0 \land x_{1} < x_{2} \Rightarrow {\begin{cases} f (x) < 0; x < x_{1} \lor x > x_{2} \\ f (x) = 0; x = x_{1} \lor x = x_{2} \\ f (x) > 0; x_{1} < x < x_{2} \end{cases}

a < 0 \land b^{2} - 4 a c = 0 \Rightarrow {\begin{cases} f (x) < 0; x \neq x_{1} \\ f (x) = 0; x = x_{1} \end{cases}

a < 0 \land b^{2} - 4 a c < 0 \Rightarrow f (x) < 0

a = 0 \land b < 0 \Rightarrow {\begin{cases} f (x) < 0; x > x_{1} \\ f (x) = 0; x = x_{1} \\ f (x) > 0; x < x_{1} \end{cases}

a = 0 \land b = 0 \Rightarrow {\begin{cases} f (x) < 0; c < 0 \\ f (x) = 0; c = 0 \\ f (x) > 0; c > 0 \end{cases}

a = 0 \land b > 0 \Rightarrow {\begin{cases} f (x) < 0; x < x_{1} \\ f (x) = 0; x = x_{1} \\ f (x) > 0; x > x_{1} \end{cases}

a > 0 \land b^{2} - 4 a c > 0 \land x_{1} < x_{2} \Rightarrow {\begin{cases} f (x) < 0; x_{1} < x < x_{2} \\ f (x) = 0; x = x_{1} \lor x = x_{2} \\ f (x) > 0; x < x_{1} \lor x > x_{2} \end{cases}

a > 0 \land b^{2} - 4 a c = 0 \Rightarrow {\begin{cases} f (x) = 0; x = x_{1} \\ f (x) > 0; x \neq x_{1} \end{cases}

a > 0 \land b^{2} - 4 a c < 0 \Rightarrow f (x) > 0

Commented by Tawa11 last updated on 14/Oct/21

Wow, I really appreciate your time sir . God bless you sir .

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