Ontario Mathematics Proficiency Practice Test

The quadratic formula is derived from the standard form of a quadratic equation, which is expressed as \( ax^2 + bx + c = 0 \). This formula provides a way to find the values of \( x \) (the roots) that satisfy the equation.

The correct formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here’s how this formula works:

- The term \( -b \) represents the opposite of the coefficient of \( x \) in the equation.

- The discriminant \( b^2 - 4ac \) determines the nature of the roots. If the result is positive, there are two distinct real roots; if it is zero, there is exactly one real root (a repeated root); if it is negative, the roots are complex.

- The square root of the discriminant, \( \sqrt{b^2 - 4ac} \), allows us to calculate the possible values for \( x \) based on whether we add or subtract it.

- Finally, dividing the entire expression by \( 2a \) ensures proper scaling of the results based on the leading coefficient of the quadratic.

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