Solving Quadratic Equations (AI SL)
Some quadratic questions don't ask you to graph or optimise anything - they ask you to solve the equation itself, or to work out what makes it solvable in the first place. This page covers the quadratic formula and the discriminant, the tool that tells you how many real roots an equation has before you even solve it. It's part of the broader Quadratic Models topic.
11 questions on this sub-topic.
The quadratic formula and the discriminant
Covered under IB syllabus reference SL2.6: modelling skills - given a context, recognise and choose an appropriate model, find the parameters of a model by solving equations from given conditions, and use technology to fit and test the model.
General quadratic model
\(f(x)=ax^2+bx+c\)
Not in the formula booklet - it's just the standard form you rearrange an equation into before solving.
Quadratic formula, for roots
\(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)
In the formula booklet, so you don't need to memorise it. The part under the root, \(b^2-4ac\), is the discriminant - it tells you how many real solutions to expect.
Need the full syllabus table and GDC walkthroughs? See Quadratic Models.
Worked examples
\(x^2 + kx + 9 = 0\) has equal roots.
(a)(i) Find the value of \(k<0\).
(a)(ii) Find the value of \(k>0.\)
Worked solution
A quadratic \(ax^2+bx+c=0\) has equal (repeated) roots exactly when its discriminant \(\Delta=b^2-4ac=0\): geometrically the parabola just touches the \(x\)-axis. M1
Here \(a=1,\ b=k,\ c=9.\) A1
\(k^2-4(1)(9)=0\Rightarrow k^2=36\Rightarrow k=\pm6.\) A1 A1
Find the values of \(k\) for which \(x^2 + kx + 4 = 0\) has two distinct real roots.
Worked solution
Need \(\Delta=b^2-4ac>0.\) M1
\(a=1,\ b=k,\ c=4\): \(k^2-16>0.\) A1
\(k^2>16\Rightarrow k<-4\) or \(k>4.\) A1 A1
Find the zeros of \(f(x)=2x^2-7x-4\) using your GDC.
Worked solution
Plot \(y=2x^2-7x-4\) on the GDC. M1
The graph crosses the \(x\)-axis at \(x=-0.5\) and \(x=4.\) A1 A1
Common mistakes
- Mixing up the discriminant conditions. \(\Delta>0\) gives two distinct real roots, \(\Delta=0\) gives one repeated root, and \(\Delta<0\) gives no real roots at all - it's easy to swap "equal roots" and "two distinct roots" under pressure.
- Dropping a solution branch. An inequality like \(k^2>16\) has two separate solution ranges, \(k<-4\) and \(k>4\) - stopping after finding just one loses marks even if the working is otherwise correct.
- Misidentifying \(a\), \(b\), \(c\) when the leading coefficient isn't 1. In \(kx^2+4x+2=0\), it's \(k\) (not \(1\)) that plays the role of \(a\) in \(b^2-4ac\) - substituting the wrong coefficient into the formula is a common slip.
Ready to practise properly?
10 solving-quadratic-equations questions, marked instantly like the real exam.
Quick answers
What does the discriminant tell you about a quadratic equation?
The discriminant \(b^2-4ac\) tells you how many real roots \(ax^2+bx+c=0\) has: two distinct real roots if it's positive, one repeated (equal) root if it's zero, and no real roots if it's negative.
Is the quadratic formula in the formula booklet?
Yes. \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) is given in the formula booklet, so you don't need to memorise it - just substitute the coefficients correctly.