Solving Quadratic Equations (DP IB Analysis & Approaches (AA): HL): Revision Note

Solving quadratic equations

How do I decide the best method to solve a quadratic equation?

• A quadratic equation is of the form $a{x}^2+bx+c=0$

• If it is a calculator paper then use your GDC to solve the quadratic

• If it is a non-calculator paper then

  • you can always use the quadratic formula

  • you can factorise if it can be factorised with integers

  • you can always complete the square

How do I solve a quadratic equation by the quadratic formula?

• If necessary rewrite in the form $a{x}^2+bx+c=0$

• Identify the values of $a$, $b$, and $c$

• Substitute the values into the formula

  • $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ 

  • This is given in the formula booklet

• Simplify the solutions if possible

How do I solve a quadratic equation by factorising?

• Factorise to rewrite the quadratic equation in the form $a(x-p)(x-q)=0$

• Equate each factor to zero and solve

  • $x-p=0\Rightarrow x=p$

  • $x-q=0\Rightarrow x=q$

How do I solve a quadratic equation by completing the square?

• Complete the square to rewrite the equation in the form $a{\left(x-h\right)}^2+k=0$

• Isolate the squared term

  • ${\left(x-h\right)}^2=-\frac{k}{a}$

• If $\left(-\frac{k}{a}\right)$ is negative then there will be no solutions

• If $\left(-\frac{k}{a}\right)$ is positive then there will be two values for $\left(x-h\right)$

  • $x-h=\pm \sqrt{-\frac{k}{a}}$

• Solve for $x$

  • $x=h\pm \sqrt{-\frac{k}{a}}$