What is the meaning for solving a quadratic equation?

Answer 1

A quadratic equation is an equation in which one variable (usually y) is in terms of another (usually x) where the highest power of that other is 2 (i.e. "x squared" or x2). Solving such an equation refers to the process of determining which values of the x-ordinate give a y-ordinate of zero.

There are technically always two roots (the aforementioned x-ordinates): either two different roots, two equal roots (where the same number is used twice) or no real roots but two imaginary/complex roots (in this case the y-ordinate never equals zero; 'imaginary'/'complex' means a number involving the square root of a negative number).

They can be calculated using a formula which involves the different constants of the equation, where y = ax2 + bx + c: the solutions are x = [-b + √(b2 - 4ac)]/2a.

For example, say an equation is y = 1x2 + 7x + 12. The corresponding formula is

x = [-7 + √(49 - 48)]/2

= (-7 + 1)/2

= -3 or -4

We know these are roots, but we should still check them by plugging each one into the initial equation. If they work, then the x-intercepts are (-3, 0) and (-4, 0).