Quadratic Formula
The quadratic formula solves any equation of the form ax^2 + bx + c = 0 for x, even when the expression will not factorise neatly. It states x = (-b plus or minus the square root of (b^2 - 4ac)) all divided by 2a.
Method
- Rearrange the equation so it equals zero, in the form ax^2 + bx + c = 0.
- Write down the values of a, b and c, being careful with negative signs.
- Substitute a, b and c into the formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a).
- Work out b^2 - 4ac first - this is called the discriminant. If it's negative, there are no real solutions.
- Find both answers: one using +sqrt(...) and one using -sqrt(...), then simplify or round as instructed.
Worked example
Solve 2x^2 + 3x - 5 = 0 using the quadratic formula.
- Here a = 2, b = 3 and c = -5.
- Substitute into the formula: x = (-3 +/- sqrt(3^2 - 4x2x(-5))) / (2x2).
- Work out the discriminant: 3^2 - 4x2x(-5) = 9 + 40 = 49.
- sqrt(49) = 7, so x = (-3 +/- 7) / 4.
- The two solutions are x = (-3 + 7)/4 = 1 and x = (-3 - 7)/4 = -2.5.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
Solve 5x^2 + 9x - 2 = 0 using the quadratic formula. Give your answers to 3 significant figures.
A rectangle has length (x + 3) cm and width x cm. Its area is 40 cm^2. Form an equation in x and use the quadratic formula to find the width of the rectangle.
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