Solving Quadratic Equations: The Quadratic Formula

The Quadratic Formula

What do you do when a quadratic equation stubbornly refuses to factorise into neat brackets? You use the quadratic formula, which calculates the exact roots (or solutions) for any solvable quadratic equation. Before using the formula, the equation must be written in standard form, meaning it is set equal to zero:

ax^2 + bx + c = 0

The letters $a$ , $b$ , and $c$ represent the coefficients and the constant in the equation. The formula is:

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

(Note: Edexcel provides this on the exam aid sheet for 2024 and 2025, but you should prepare to memorise it for 2026 onwards).

Using the Discriminant

The expression under the square root, $b^2 - 4ac$ , is called the discriminant. It acts as a quick test to determine the nature of the real roots before you calculate them fully:

$b^2 - 4ac > 0$ : There are two distinct real solutions (the graph crosses the x-axis twice).
$b^2 - 4ac = 0$ : There is one repeated root or equal root (the graph's turning point touches the x-axis at one point).
$b^2 - 4ac < 0$ : There are no real solutions (your calculator will show a "Math Error").

Rearranging and Solving

Examiners will often ask for answers to a specified degree of accuracy, such as "to 2 decimal places". This is a massive hint that the equation will not factorise and you must use the formula. Always ensure your equation is set to zero before identifying $a$ , $b$ , and $c$ . If terms are on the right-hand side, move them over by changing their sign. Sometimes you will be asked to leave your answer in surd form. This means simplifying the square root and leaving the answer as an exact fraction, rather than a rounded decimal.

Worked Example

Solve $3x^2 + 2x = 4$ . Give your solutions in exact surd form and to 2 decimal places.

Step 1: Rearrange the equation into standard form ( $ax^2 + bx + c = 0$ ).

$3x^2 + 2x - 4 = 0$

Step 2: State the quadratic formula and identify the coefficients.

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$a = 3$ , $b = 2$ , $c = -4$

Step 3: Substitute the values into the formula.

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

Step 4: Simplify to find the exact surd form.

$x = \frac{-2 \pm \sqrt{4 + 48}}{6}$
$x = \frac{-2 \pm \sqrt{52}}{6}$
Simplify the surd: $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$
$x = \frac{-2 \pm 2\sqrt{13}}{6} = \frac{-1 \pm \sqrt{13}}{3}$

Step 5: Calculate the final decimal answers to 2 decimal places.

$x = 0.87$ or $x = -1.54$

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Exam Tips

Common Mistake
Students often extend the fraction line only under the square root; remember that the entire numerator $(-b \pm \sqrt{b^2 - 4ac})$ must be divided by $2a$ .
Common Mistake
Failing to use brackets when squaring a negative $b$ value on a calculator will result in an incorrect negative discriminant (e.g., typing $-3^2$ gives $-9$ , whereas $(-3)^2$ correctly gives $9$ ).
3
In Edexcel exams, if a question asks for solutions to a specific degree of accuracy like '2 decimal places', it is a massive hint that the equation will not factorise and you must use the quadratic formula.
4
To gain the first method mark (M1), always write out the full substitution of $a$ , $b$ , and $c$ into the formula clearly before reaching for your calculator.

Key Terms(9)

Quadratic formula: A mathematical formula used to find the exact roots of any solvable quadratic equation.
Standard form: A quadratic equation written as $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Roots: The values of $x$ that make the equation equal to zero, representing where the graph crosses the x-axis.
Coefficients: The numerical values multiplied by a variable, such as $a$ and $b$ in a quadratic equation.
Constant: The term in an algebraic expression that does not contain any variables, represented by $c$ in a quadratic equation.
Discriminant: The expression $b^2 - 4ac$ found under the square root in the quadratic formula, used to determine the nature of the roots.
Real roots: The $x$ -values where the curve intersects the $x$ -axis, which exist when the discriminant is greater than or equal to zero.
Repeated root: A single solution that occurs when the discriminant equals zero, meaning the quadratic is a perfect square.
Surd form: An exact numerical answer left containing a square root symbol rather than being evaluated as a decimal.

Previous NoteSolving Quadratic Equations: Completing the Square Next NoteSolving Quadratic Equations: Graphical Methods

Back to Quadratic Equations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Quadratic formula: A mathematical formula used to find the exact roots of any solvable quadratic equation.
Standard form: A quadratic equation written as $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Roots: The values of $x$ that make the equation equal to zero, representing where the graph crosses the x-axis.
Coefficients: The numerical values multiplied by a variable, such as $a$ and $b$ in a quadratic equation.
Constant: The term in an algebraic expression that does not contain any variables, represented by $c$ in a quadratic equation.
Discriminant: The expression $b^2 - 4ac$ found under the square root in the quadratic formula, used to determine the nature of the roots.
Real roots: The $x$ -values where the curve intersects the $x$ -axis, which exist when the discriminant is greater than or equal to zero.
Repeated root: A single solution that occurs when the discriminant equals zero, meaning the quadratic is a perfect square.
Surd form: An exact numerical answer left containing a square root symbol rather than being evaluated as a decimal.

Solving Quadratic Equations: The Quadratic Formula

The Quadratic Formula

ax^2 + bx + c = 0

The letters $a$ , $b$ , and $c$ represent the coefficients and the constant in the equation. The formula is:

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

(Note: Edexcel provides this on the exam aid sheet for 2024 and 2025, but you should prepare to memorise it for 2026 onwards).

Using the Discriminant

The expression under the square root, $b^2 - 4ac$ , is called the discriminant. It acts as a quick test to determine the nature of the real roots before you calculate them fully:

$b^2 - 4ac > 0$ : There are two distinct real solutions (the graph crosses the x-axis twice).
$b^2 - 4ac = 0$ : There is one repeated root or equal root (the graph's turning point touches the x-axis at one point).
$b^2 - 4ac < 0$ : There are no real solutions (your calculator will show a "Math Error").

Rearranging and Solving

Worked Example

Solve $3x^2 + 2x = 4$ . Give your solutions in exact surd form and to 2 decimal places.

Step 1: Rearrange the equation into standard form ( $ax^2 + bx + c = 0$ ).

$3x^2 + 2x - 4 = 0$

Step 2: State the quadratic formula and identify the coefficients.

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$a = 3$ , $b = 2$ , $c = -4$

Step 3: Substitute the values into the formula.

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

Step 4: Simplify to find the exact surd form.

$x = \frac{-2 \pm \sqrt{4 + 48}}{6}$
$x = \frac{-2 \pm \sqrt{52}}{6}$
Simplify the surd: $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$
$x = \frac{-2 \pm 2\sqrt{13}}{6} = \frac{-1 \pm \sqrt{13}}{3}$

Step 5: Calculate the final decimal answers to 2 decimal places.

$x = 0.87$ or $x = -1.54$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often extend the fraction line only under the square root; remember that the entire numerator $(-b \pm \sqrt{b^2 - 4ac})$ must be divided by $2a$ .
Common Mistake
Failing to use brackets when squaring a negative $b$ value on a calculator will result in an incorrect negative discriminant (e.g., typing $-3^2$ gives $-9$ , whereas $(-3)^2$ correctly gives $9$ ).
3
In Edexcel exams, if a question asks for solutions to a specific degree of accuracy like '2 decimal places', it is a massive hint that the equation will not factorise and you must use the quadratic formula.
4
To gain the first method mark (M1), always write out the full substitution of $a$ , $b$ , and $c$ into the formula clearly before reaching for your calculator.

Key Terms(9)

Quadratic formula: A mathematical formula used to find the exact roots of any solvable quadratic equation.
Standard form: A quadratic equation written as $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Roots: The values of $x$ that make the equation equal to zero, representing where the graph crosses the x-axis.
Coefficients: The numerical values multiplied by a variable, such as $a$ and $b$ in a quadratic equation.
Constant: The term in an algebraic expression that does not contain any variables, represented by $c$ in a quadratic equation.
Discriminant: The expression $b^2 - 4ac$ found under the square root in the quadratic formula, used to determine the nature of the roots.
Real roots: The $x$ -values where the curve intersects the $x$ -axis, which exist when the discriminant is greater than or equal to zero.
Repeated root: A single solution that occurs when the discriminant equals zero, meaning the quadratic is a perfect square.
Surd form: An exact numerical answer left containing a square root symbol rather than being evaluated as a decimal.

Previous NoteSolving Quadratic Equations: Completing the Square Next NoteSolving Quadratic Equations: Graphical Methods

Back to Quadratic Equations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Quadratic formula: A mathematical formula used to find the exact roots of any solvable quadratic equation.
Standard form: A quadratic equation written as $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .
Roots: The values of $x$ that make the equation equal to zero, representing where the graph crosses the x-axis.
Coefficients: The numerical values multiplied by a variable, such as $a$ and $b$ in a quadratic equation.
Constant: The term in an algebraic expression that does not contain any variables, represented by $c$ in a quadratic equation.
Discriminant: The expression $b^2 - 4ac$ found under the square root in the quadratic formula, used to determine the nature of the roots.
Real roots: The $x$ -values where the curve intersects the $x$ -axis, which exist when the discriminant is greater than or equal to zero.
Repeated root: A single solution that occurs when the discriminant equals zero, meaning the quadratic is a perfect square.
Surd form: An exact numerical answer left containing a square root symbol rather than being evaluated as a decimal.