Quadratic equations

From GMATClub

The general form of quadratic equations is <m>ax^2 + bx + c = 0</m>. Depending on the coefficients, the equation may have zero, one, or two real solutions for <m>x</m>:

<m>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}</m>

The part with the square root is called the discriminant:

<m>D=\sqrt{b^2-4ac}</m>

Calculating the discriminant is sufficient to know how many solutions does the equation have: for positive <m>D</m> there are two solutions, if <m>D=0</m> there is only one, and for <m>D<0</m> there are none.

Viète's theorem

If <m>x1</m> and <m>x2</m> are the roots of quadratic equation <m>ax^2 + bx + c = 0</m>, then:

• <m>x1+x2=-\frac{b}{a}</m>
• <m>x1*x2=\frac{c}{a}</m>

Special consideration should be given to quadratic equation when <m>a=1</m>. Thus, it takes this form: <m>x^2 + px + q = 0</m>. In this case the roots can be found following the formulae:

• <m>x1+x2=-p</m>
• <m>x1*x2=q</m>

For example:

<m>x^2-3x-10=0</m>

We see that <m>q=-10</m> and <m>-p=3</m>.

We know from the above formulae that <m>x1*x2=-10</m> and <m>x1+x2=3</m>.

It can be seen that the roots are -2 and 5. You need to solve the system of equations if you cannot see the roots at once.

Knowing these properties of roots of quadratic equation can greatly speed up solving the equations like <m>x^2 + px + q = 0</m>, thus saving precious time on GMAT.