Find the interval(s) for b such that the equation has at least one real solution and write a conjecture about the interval(s) based on the values of the coefficients $4 x ^ { 2 } + b x + 7 = 0$ . A) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a 0$ and $c 0$ and $c 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ D) $( - \infty , 2 \sqrt { 28 } ] \cup [ 2 \sqrt { 28 } , \infty )$ If $a 0$ and $c 0$ , $b \leq 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ E) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a

Question

Find the interval(s) for b such that the equation has at least one real solution and write a conjecture about the interval(s) based on the values of the coefficients $4 x ^ { 2 } + b x + 7 = 0$ . A) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c < 0$ , $b \geq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ B) $( - \infty , - 2 \sqrt { 28 } ] \cup [ - 2 \sqrt { 28 } , \infty )$ If $a < 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$ C) $( - \infty , - 2 \sqrt { 28 } ] \cup [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ D) $( - \infty , 2 \sqrt { 28 } ] \cup [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$ E) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a < 0$ and $c < 0$ , $b \leq 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$

Quizplus · Accepted Answer

For a quadratic equation $ax^2 + bx + c = 0$ to have at least one real solution, the discriminant $b^2 - 4ac \geq 0$. Here, $a = 4$ and $c = 7$, both positive. The condition becomes $b^2 \geq 4 \cdot 4 \cdot 7$, or $b^2 \geq 112$, which simplifies to $b \leq -2\sqrt{28}$ or $b \geq 2\sqrt{28}$, corresponding to choice C.

Find the Interval(s) for B Such That the Equation Has 4x2+bx+7=04 x ^ { 2 } + b x + 7 = 04x2+bx+7=0

Find the Interval(s) for B Such That the Equation Has $4 x ^ { 2 } + b x + 7 = 0$