Solving Quadratic Equations: Quadratic Formula
The quadratic formula solves any equation written as ax² + bx + c = 0, even when it won't factorise. Put a, b and c into x = (−b ± √(b² − 4ac)) ÷ 2a to find the roots. Its discriminant, b² − 4ac, gives the number of solutions.
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The Quadratic Formula and the Discriminant
- For any quadratic in the form , the solutions are given by .
- The part under the square root, , is called the discriminant and tells you how many solutions there are.
- If the discriminant is negative, there are no real solutions, so you can stop calculating.
Example: Using the Quadratic Formula
- For , substitute , , and into the formula.
- The discriminant is , which means there are two different real solutions.
- Substituting gives , so the solutions are and .
Example: Rearranging Before Using the Formula
- First rearrange into the form : .
- Simplify by dividing every term by 2 to get , which makes the calculations much easier.
- Substitute , , and into the quadratic formula to find the solutions.
Practice Questions
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What is the quadratic formula?
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The correct quadratic formula is . It is used to find the solutions (roots) of a quadratic equation.
What does a negative discriminant indicate about the roots of the quadratic equation?
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A negative discriminant means there are no real roots. If it is zero, there is one real root, and if it is positive, there are two distinct real roots.
What is the discriminant of the quadratic equation ?
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The discriminant of a quadratic equation is . For , , , and . The discriminant is , which indicates one real root.
For the equation , what are the roots?
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Using the quadratic formula with , , and , the discriminant is . The roots are , which gives and .
For the equation , what are the roots?
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Using the quadratic formula with , , and , the discriminant is . The roots are , which gives and .
For the equation , what are the roots?
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Using the quadratic formula with , , and , the discriminant is . The roots are , which gives and .
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Students Also Ask
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The discriminant, b² - 4ac, tells you how many real roots a quadratic equation has before you solve it. A positive discriminant gives two distinct real roots. A discriminant of zero gives exactly one real root. A negative discriminant means there are no real roots.
A negative discriminant means b² - 4ac is less than zero, so the quadratic equation has no real roots. Because you would need the square root of a negative number, there are no real solutions to find. You can stop calculating straight away.
A discriminant of zero means b² - 4ac equals zero, so the quadratic equation has exactly one real root. Adding or subtracting the square root of zero gives the same value. This means the plus and minus parts of the formula give a single repeated solution.
You can use the quadratic formula to solve any quadratic equation written in the form ax² + bx + c = 0. It works for every quadratic once you have identified the values of a, b, and c. This makes it reliable even when an equation looks complicated.
Yes. Before identifying a, b, and c, move all the terms to one side. The equation must be in the form ax² + bx + c = 0. Simplifying first, such as dividing every term by a common factor, makes the calculations much easier.