Find the zeros, if any, of the quadratic function using the quadratic formula. What are the \( x \)-intercepts, if any, of the graph of the function? \( g(x)=5 x^{2}+7 x+2 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) A. The zeros and the \( x \)-intercepts are the same. They are B. The zeros and the \( x \)-intercepts are different. The zeros are C. There is no real zero solution and no \( x \)-intercept.

To find the zeros of the quadratic function \( g(x) = 5x^2 + 7x + 2 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 5 \), \( b = 7 \), and \( c = 2 \). First, we calculate the discriminant: \[ b^2 - 4ac = 7^2 - 4(5)(2) = 49 - 40 = 9 \] Since the discriminant is positive, we have two real solutions. Next, we substitute into the quadratic formula: \[ x = \frac{-7 \pm \sqrt{9}}{10} = \frac{-7 \pm 3}{10} \] Calculating the two potential \( x \)-values: 1. \( x = \frac{-7 + 3}{10} = \frac{-4}{10} = -\frac{2}{5} \) 2. \( x = \frac{-7 - 3}{10} = \frac{-10}{10} = -1 \) Thus, the zeros of the function and the \( x \)-intercepts of the graph are: \[ -\frac{2}{5}, -1 \] So, the correct choice is: A. The zeros and the \( x \)-intercepts are the same. They are \( -\frac{2}{5}, -1 \). --- Want to know a fun fact? The quadratic formula is like a magical treasure map! It guides you straight to the roots of any quadratic function, ensuring you never get lost in the vast array of numbers. Unlike pi, which is infinite and non-repeating, the roots you find using this formula will always be tidy if the discriminant is a perfect square! If you're looking to avoid common pitfalls with quadratics, remember to double-check your discriminant. A negative result means no real roots, so no intercepts! Steering clear of calculation errors ensures you hit the target every time!