Find the interval(s) for b such that the following equation has at least one real solution and write a conjecture about the interval(s) based on the values of the coefficients $4 x ^ { 2 } + b x + 13 = 0$ . A) $( - \infty , - 2 \sqrt { 52 } ] \cup [ 2 \sqrt { 52 } , \infty )$ If $a 0$ and $c 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ B) $( - \infty , - 2 \sqrt { 52 } ] \cup [ - 2 \sqrt { 52 } , \infty )$ If $a 0$ and $c 0$ and $c 0$ , $- b \leq - 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$ D) $( - \infty , 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $a 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ E) $( - \infty , - 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $a

Question

Find the interval(s) for b such that the following equation has at least one real solution and write a conjecture about the interval(s) based on the values of the coefficients $4 x ^ { 2 } + b x + 13 = 0$ . A) $( - \infty , - 2 \sqrt { 52 } ] \cup [ 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ B) $( - \infty , - 2 \sqrt { 52 } ] \cup [ - 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c < 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ C) $( - \infty , 2 \sqrt { 52 } ] \cup [ 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c > 0$ , $- b \leq - 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$ D) $( - \infty , 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $a < 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ E) $( - \infty , - 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $a < 0$ and $c < 0$ , $b \leq 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$

Quizplus · Accepted Answer

For a quadratic equation $ax^2 + bx + c = 0$ to have at least one real solution, the discriminant $b^2 - 4ac$ must be greater than or equal to 0. Here, $a = 4$ and $c = 13$, so the condition becomes $b^2 - 4(4)(13) \geq 0$, which simplifies to $b^2 \geq 208$. This means $b \leq -2\sqrt{52}$ or $b \geq 2\sqrt{52}$, matching the interval in choice A and the conjecture for $a > 0$ and $c > 0$.

Find the Interval(s) for B Such That the Following Equation 4x2+bx+13=04 x ^ { 2 } + b x + 13 = 04x2+bx+13=0

Find the Interval(s) for B Such That the Following Equation $4 x ^ { 2 } + b x + 13 = 0$