[Algebra 9] Quadratic equation in one variable

❄ QUADRATIC EQUATION IN ONE VARIABLE

The standard form

${a{x^2} + bx + c = 0}$

where x is unknown; a, b and c are real numbers with $a \ne 0$.

General quadratic equation

Calculate $\Delta  = {b^2} - 4ac$

$\Delta  < 0$ $ \Rightarrow $ the equation has no solution

$\Delta  = 0$ $ \Rightarrow $ the equation has a double solution: ${x_{1,2}} = \dfrac{{ - b}}{{2a}}$

$\Delta  > 0$ $ \Rightarrow $ the equation has two distinct solutions: ${x_{1,2}} = \dfrac{{ - b \pm \sqrt \Delta  }}{{2a}}$

+ In the cases when the coefficient b is an even number

* In special cases:

If $b = 0$ or $c = 0$, we convert the quadratic equation in to the product equation.
$a + b + c = 0$ $ \Rightarrow $ ${x_1} = 1$, ${x_2} = \dfrac{c}{a}$
$a - b + c = 0$ $ \Rightarrow $ ${x_1} = -1$, ${x_2} = \dfrac{-c}{a}$

Example. Solve the following equations:
1) ${x^2} + 4x = 0$
eqn $ \Leftrightarrow x\left( {x + 4} \right) = 0$
$ \Leftrightarrow x = 0\;or\;x + 4 = 0$
$ \Leftrightarrow x = 0\;or\;x = -4$
The equation has two solutions $x = 0,\;x =  -4$.
2) ${x^2} - 9 = 0$
eqn $ \Leftrightarrow \left( {x - 3} \right)\left( {x + 3} \right) = 0$
$ \Leftrightarrow x - 3 = 0\;or\;x + 3 = 0$
$ \Leftrightarrow x = 3\;or\;x = -3$
The equation has two solutions $x = 3,\;x =  -3$.
3) ${x^2} +1 = 0$
We have:
${x^2} \ge 0$ for all $x \in$ $\mathbb{R}$
$ \Rightarrow {x^2} + 1 \ge 1 > 0$
Therefore, the equation has no solution.
4) ${{x^2} - 5x + 6 = 0}$

$\left( {a = 1,\;b =  - 5,\;c = 6} \right)$

$\Delta  = {\left( { - 5} \right)^2} - 4.1.6 = 25 - 24 = 1 > 0$

The equation has two distinct solutions:

${x_1} = \dfrac{{ - \left( { - 5} \right) - \sqrt 1 }}{{2.1}} = 2$, ${x_2} = \dfrac{{ - \left( { - 5} \right) + \sqrt 1 }}{{2.1}} = 3$.

5) ${x^2} - \sqrt 5 x + \dfrac{5}{4} = 0$

$\left( {a = 1,\;b =  - \sqrt 5 ,\;c = \dfrac{5}{4}} \right)$

$\Delta  = {\left( { - \sqrt 5 } \right)^2} - 4.1.\dfrac{5}{4} = 5 - 5 = 0$

The equation has a double solution: $x = \dfrac{{ - \left( { - \sqrt 5 } \right)}}{{2.1}} = \dfrac{{\sqrt 5 }}{2}$.

6) ${2{x^2} + 3x + 2 = 0}$

$\left( {a = 2,\;b =  3,\;c = 2} \right)$

$\Delta  = {3^2} - 4.2.2 = 9 - 16 =  - 7 < 0$
The equation has no solution.
7) ${{x^2} - 3x + 2 = 0}$
$\left( {a = 1,\;b =  -3,\;c = 2} \right)$
The equation of the form $a + b + c = 0$
$ \Rightarrow $ The equation has two solutions ${x_1} = 1$, ${x_2} = \dfrac{2}{1} = 2$
8) ${{x^2} - 4x - 5 = 0}$
$\left( {a = 1,\;b =  -4,\;c = -5} \right)$
The equation of the form $a - b + c = 0$
$ \Rightarrow $ The equation has two solutions ${x_1} = -1$, ${x_2} = \dfrac{{ - \left( { - 5} \right)}}{1} = 5$