Mathematics \( 1.1 .2 \quad 5 x^{2}+10 x-3 \)

Did you know that mathematicians have been solving quadratic equations like the one you've presented since ancient times? The methods we use today—like completing the square or the quadratic formula—were developed over centuries, with contributions from Indian, Persian, and later European scholars. This particular quadratic, formed of a coefficient in front of \(x^2\) that influences its "shape," is a delightful example of how algebra can represent various scenarios, from physics to finance! When tackling this quadratic equation, a common mistake is neglecting the discriminant, given by \(b^2 - 4ac\). If the discriminant is negative, the equation has no real solutions (only complex ones), which can be a surprise! Always check this part! Also, remember to factor when possible or simplify your expressions before using the quadratic formula, as it can save time and reduce errors.