We have an answer to "The equation 2x2+ax+(b+3)=0 has real roots. Find the..."

Leroy Davidson

Answered question

2022-04-02

The equation

2 x^{2} + a x + (b + 3) = 0

has real roots. Find the minimum value of

a^{2} + b^{2}

Ben Castillo

Beginner2022-04-03Added 13 answers

By your work:

a^{2} + b^{2} = a^{2} + b^{2} - \frac{3}{4} (a^{2} - 8 (b + 3)) + \frac{3}{4} (a^{2} - 8 (b + 3)) \geq

\geq a^{2} + b^{2} - \frac{3}{4} (a^{2} - 8 (b + 3)) = \frac{1}{4} (a^{2} + 4 {(b + 3)}^{2}) + 9 \geq 9 .

The equality occurs for

a = 0 and b = - 3

, which says that we got a minimal value.

Marquis Ibarra

Beginner2022-04-04Added 9 answers

The minimum value for

a^{2} + b^{2} = 8 (b + 3) + b^{2}

, because

a^{2} \geq q 8 (b + 3)

.
The value of b that minimizes the quadratic

8 (b + 3) + b^{2}

is:

8 + 2 b = 0 \Rightarrow b = - 4

From

a^{2} \geq q 8 (b + 3) \Rightarrow b \geq q - 3

Therefore, the minimum value of

a^{2} + b^{2}

is when

b = - 3

and

a = 0

\Rightarrow a^{2} + b^{2} \geq q {(0)}^{2} + {(- 3)}^{2}

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