Quadratic formulas

Algebra Level 3

The quadratic formula for the quadratic equation that a x 2 + b x + c = 0 ax^{2} + bx + c = 0 is − b ± b 2 − 4 a c 2 a \displaystyle \frac { -b \pm \sqrt { b^{2} - 4ac } } { 2a } .

But it is only the formula when the b b is odd.

Then what is the formula when the b b is even?

So a x 2 + 2 b x + c = 0 ax^{2} + 2bx + c = 0 .

− b ± b 2 ± a c a \frac { -b \pm \sqrt {b^{2} \pm ac } } { a } − b ± b 2 − 4 a c a \frac { -b \pm \sqrt { b^{2} - 4ac } } { a } − b ± b 2 ± 4 a c a \frac { -b \pm \sqrt { b^{2} \pm 4ac } } { a } − b ± b 2 + a c a \frac { -b \pm \sqrt { b^{2} + ac } } { a } − b ± b 2 − a c a \frac { -b \pm \sqrt { b^{2} - ac } } { a } − b ± b 2 + 4 a c a \frac { -b \pm \sqrt { b^{2} + 4ac } } { a }

. . by . . , 14, South Korea

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1 solution

. .
Feb 15, 2021

The quadratic formula for the quadratic equation that a x 2 + b x + c = 0 ax^{2} + bx + c = 0 when b b is odd is − b ± b 2 − 4 a c 2 a \displaystyle \frac { -b \pm \sqrt { b^{2} - 4ac } } { 2a } .

Because a x 2 + b x + c = 0 → x 2 + b a x + c a = 0 → x 2 + b a x = − c a → x 2 + b a + ( b 2 a ) 2 = − c a + b 2 4 a 2 → ( x + b 2 a ) 2 = − 4 a c 4 a 2 + b 2 4 a 2 → x + b 2 a = ± b 2 − 4 a c 2 a → x = − b ± b 2 − 4 a c 2 a \displaystyle ax^{2} + bx + c = 0 \rightarrow x^{2} + \frac{b}{a}x + \frac{c}{a} = 0 \rightarrow x^{2} + \frac{b}{a}x = - \frac{c}{a} \rightarrow x^{2} + \frac{b}{a} + \left(\frac{b}{2a}\right)^{2} = - \frac{c}{a} + \frac{b^{2}}{4a^{2}} \rightarrow \left(x+\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^{2}}{4a^{2}} \rightarrow x+\frac{b}{2a} = \pm \frac{ \sqrt{ b^{2} - 4ac } } { 2a } \rightarrow x = \frac { -b \pm \sqrt { b^{2} - 4ac } } { 2a } .

And if the quadratic equation that a x 2 + 2 b x + c = 0 ax^{2} + 2bx + c = 0 , the formula is − b ± b 2 − a c a \displaystyle \frac { -b \pm \sqrt { b^{2} - ac } } { a } .

Because a x 2 + 2 b x + c = 0 → x 2 + 2 b a x + c a = 0 → x 2 + 2 b a x = − c a → x 2 + 2 b a + ( b a ) 2 = − c a + b 2 a 2 → ( x + b a ) 2 = − a c + b 2 a 2 → x + b a = ± b 2 − a c a → x = − b ± b 2 − a c a \displaystyle ax^{2} + 2bx + c = 0 \rightarrow x^{2} + \frac{2b}{a}x + \frac{c}{a} = 0 \rightarrow x^{2} + \frac{2b}{a}x = - \frac{c}{a} \rightarrow x^{2} + \frac{2b}{a} + \left(\frac{b}{a}\right)^{2} = - \frac{c}{a} + \frac{b^{2}}{a^{2}} \rightarrow \left(x+\frac{b}{a}\right)^{2} = \frac{ -ac + b^{2} } { a^{2} } \rightarrow x+\frac{b}{a} = \pm \frac { \sqrt{b^{2} - ac } } { a } \rightarrow x = \frac { -b \pm \sqrt { b^{2} - ac } } { a } .

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