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A quadratic equation is a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are constants with \( a \) being nonzero. This equation is quadratic because the highest power of the variable, \( x \), is two.

The solutions to a quadratic equation are found using various methods, with the quadratic formula being one of the most comprehensive. This formula states \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and provides solutions for any form of a quadratic equation. In the quadratic formula, the terms \( -b \pm \sqrt{b^2 - 4ac} \) represent the numerator which contains two critical components: the opposite of coefficient \( b \) and the square root of the discriminant \( D \), which is \( b^2 - 4ac \).

To solve a quadratic equation like \( 5x^2 - 7x + 3 = 0 \) using the quadratic formula, one must first identify the coefficients corresponding to \( a \), \( b \) and \( c \) in the formula; these are key in determining the nature and number of solutions.

The discriminant of a quadratic equation, typically denoted as \( D \) or \( \Delta \), is a mathematical expression given by \( D = b^2 - 4ac \). It plays a vital role in understanding the nature of the roots of the equation. The value of the discriminant tells us if the solutions are real or complex, and whether they are distinct or coincide.

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one unique real solution, which is also known as a repeated or double root.
• If \( D < 0 \), the equation has two complex solutions.

In the provided example, calculating the discriminant with values \( a = 5 \) , \( b = -7 \) , and \( c = 3 \) yields \( D = (-7)^2 - 4 \cdot 5 \cdot 3 \) which simplifies to \( D = 49 - 60 = -11 \). A negative discriminant indicates that the solutions to the equation will be complex numbers since the square root of a negative number is not a real number.

Complex solutions arise when the discriminant of a quadratic equation is less than zero, leading to the square root of a negative number. A complex number has a real part and an imaginary part and can be expressed as \( a + bi \), where \( i \), called the imaginary unit, is defined as \( \sqrt{-1} \).

When the discriminant is negative, the quadratic formula yields solutions in the form of complex numbers, specifically \( x = \frac{-b \pm i\sqrt{|D|}}{2a} \) where \( |D| \) is the absolute value of the discriminant. For our example, with \( D = -11 \), we find the square root of the discriminant to be \( i\sqrt{11} \).

The quadratic formula then provides us with the solutions \( x = \frac{7 \pm i\sqrt{11}}{10} \) which are both complex. Every quadratic equation has two solutions, and in the case of a negative discriminant, they are complex conjugates of each other, meaning if one solution is \( a + bi \) then the other is \( a - bi \) for same values of \( a \) and \( b \).