How to Find the Roots of a Quadratic Equation

A quadratic equation has the general form (ax^2+bx+c=0), where (a\neq0). Its roots are the values of (x) that make the equation equal to zero. They are also called solutions, zeros, or (x)-intercepts, depending on whether you are working with algebra, functions, or graphs.

Finding these values is a core skill in secondary-school mathematics. Students may meet quadratic equations in Year 10, VCE in Victoria, the HSC in New South Wales, or other senior secondary courses across Australia. The method you choose often depends on the coefficients and on whether an exact answer or a decimal approximation is required.

A quadratic can have two real roots, one repeated real root, or no real roots. These possibilities become much easier to recognise once you understand the discriminant and the shape of the corresponding parabola. Factoring, completing the square, and using the quadratic formula each reveal the solutions in a slightly different way.

CurryPlum’s step-by-step maths resources can help learners compare these methods, check algebraic working, and use calculators appropriately. This is useful when revising for an ATAR subject, completing homework after school in Brisbane or Perth, or preparing for an assessment where clear reasoning matters as much as the final answer.

Recognising A Quadratic And Its Roots

Before solving, write the equation in standard form:

[ ax^2+bx+c=0 ]

The coefficient (a) must be non-zero, because an equation without an (x^2) term is linear rather than quadratic. For example, (2x^2-7x+3=0) has (a=2), (b=-7), and (c=3). The roots are the (x)-values that satisfy this equation.

A graph gives the same information visually. If (y=ax^2+bx+c), the roots are the points where the parabola crosses or touches the (x)-axis. A graphing calculator or online graphing tool can provide a useful check, although graph estimates may be rounded. Exact algebraic solutions are generally preferred in formal working.

Solving By Factorising

Factorising is often the quickest approach when the quadratic has simple integer or rational roots. For example:

[ x^2-5x+6=0 ]

Find two numbers that multiply to (6) and add to (-5): (-2) and (-3). Therefore,

[ (x-2)(x-3)=0 ]

The zero-product property says that if two factors multiply to zero, at least one factor must be zero. Set each factor equal to zero:

[ x-2=0 \quad \text{or} \quad x-3=0 ]

So the roots are (x=2) and (x=3).

When the leading coefficient is not 1, look for factors of (ac), or use grouping. Consider (2x^2+7x+3=0). The expression factors as

[ (2x+1)(x+3)=0 ]

This gives (x=-\frac12) or (x=-3). Always expand the factors afterwards to confirm that they reproduce the original quadratic.

Using The Quadratic Formula

Some quadratics do not factor neatly. The quadratic formula works for every equation written as (ax^2+bx+c=0):

[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ]

The symbol (\pm) means that two calculations are required: one using plus and one using minus. For (3x^2-4x-2=0), identify (a=3), (b=-4), and (c=-2):

[ x=\frac{-(-4)\pm\sqrt{(-4)^2-4(3)(-2)}}{2(3)} ]

[ x=\frac{4\pm\sqrt{40}}{6} ]

Since (\sqrt{40}=2\sqrt{10}), the exact roots are

[ x=\frac{2+\sqrt{10}}{3} \quad\text{and}\quad x=\frac{2-\sqrt{10}}{3} ]

A calculator gives approximate values when a decimal answer is more practical. Enter brackets carefully, particularly around the negative value of (b). On many Australian school calculators, using a fraction template and the (\sqrt{}) key reduces input errors.

Understanding The Discriminant

The expression beneath the square root,

[ \Delta=b^2-4ac ]

is called the discriminant. It predicts the number and type of real roots before you complete the calculation. If (\Delta>0), there are two different real roots. If (\Delta=0), there is one repeated real root. If (\Delta<0), there are no real roots, although two complex roots exist.

For example, (x^2+4x+4=0) has discriminant (16-16=0). It can be written as ((x+2)^2=0), so its repeated root is (x=-2). The parabola touches the (x)-axis at this point and turns around.

For (x^2+2x+5=0), the discriminant is (4-20=-16). The graph does not meet the (x)-axis, so there are no real solutions. This interpretation is valuable in graphing and modelling questions, including problems involving area, height, or the point at which a quantity reaches zero.

Completing The Square And Checking Solutions

Completing the square changes a quadratic into vertex form. Starting with

[ x^2+6x+5=0 ]

move the constant and add the square of half the (x)-coefficient:

[ x^2+6x+9=4 ]

Thus,

[ (x+3)^2=4 ]

Taking square roots gives (x+3=\pm2), so (x=-1) or (x=-5). This method shows the turning point as well as the roots and connects directly with the vertex-form skills used in algebra and geometry.

Checking is a simple step that protects against sign mistakes. Substitute each proposed root into the original equation, rather than only into a rearranged line. For (x^2+6x+5=0), substituting (x=-1) gives (1-6+5=0), and substituting (x=-5) gives (25-30+5=0). Both values work, confirming the result.

A well-presented solution should state the method, show the key algebra, and give exact values before rounded decimals where appropriate. In an Australian classroom or assessment, working can earn marks even if a calculator entry produces an incorrect final number. Keeping exact forms such as (\frac{2\pm\sqrt{10}}{3}) also avoids unnecessary rounding.

The central process is consistent: put the quadratic equal to zero, identify (a), (b), and (c), choose factorising, completing the square, or the quadratic formula, and then verify the answers. Roots are the values that make the expression zero, while graphically they are the parabola’s intersections with the (x)-axis. That link between algebra and graphs is what the reader should remember.