Solving Simple Quadratic Equations
Solving simple quadratic equations means finding the x values that make them true. For x² = k, take the square root to get x = ±√k, so x² = 9 gives x = 3 or −3. For x² − dx = 0, factorise to x(x − d) = 0, so x = 0 or x = d.
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Solving Equations of the Form
- If , take the square root to get . Important: this only works when .
- For example, solving gives , so the solutions are and .
Example: Solving
- Rearrange by adding 12 to both sides so it becomes , which is now in the form .
- Divide both sides by 3 to simplify and get .
- Take the square root to get , so the solutions are and .
When There Is No Real Solution
- If equals a negative number, there is no real solution because no real number squares to a negative.
- For example, has no real solution, so −5 is not an answer.
Solving Equations of the Form
- Factorise to get .
- Set each factor equal to zero to get and .
Example:
- First make the coefficient of equal to 1 by dividing by 5, which helps to simplify the calculations.
- Factorise to get , so the solutions are or .
Solving Simple Quadratic Equations (Summary)
- If an equation is in the form , the solutions are and this only works when .
- If an equation is in the form , factorise to and the solutions are and .
Practice Questions
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What's the solution for ?
Correct! 🎉 +10 pointsNot quite right
To solve , take the square root of both sides. Since both 5 and −5 squared give 25, the solutions are and .
What's the solution for ?
Correct! 🎉 +10 pointsNot quite right
The only solution is , because . The value is not a solution since .
What's the solution for ?
Correct! 🎉 +20 pointsNot quite right
Since has a negative value on the right side, there are no real solutions. The square of any real number is always zero or positive.
What's the solution for ?
Correct! 🎉 +20 pointsNot quite right
First isolate to get . Taking the square root of both sides gives , so the solutions are and .
What's the solution for ?
Correct! 🎉 +20 pointsNot quite right
Isolate to get . Taking the square root gives , so the solutions are and .
What is the solution for ?
Correct! 🎉 +30 pointsNot quite right
First isolate by adding 125 to both sides to get . Then divide by 5 to get . Taking the square root gives and .
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Students Also Ask
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A simple quadratic equation in the form x² = k usually has two solutions. They are the positive and the negative square root of k. For example, x² = 9 gives x = 3 and x = -3, because squaring either value returns 9.
If a simple quadratic equation rearranges to x² equals a negative number, it has no real solutions. This is because x squared is never below zero. No real value of x can square to give a negative result, so the equation has no real answer.
A quadratic with no constant term looks like x² - dx = 0. To solve it, you factorise out the common factor x. This gives x times a bracket, equal to zero. The equation is then true when x = 0 or when the bracket equals zero. That gives the two solutions.
You should not divide both sides by x, because you would lose the solution x = 0. Dividing by x removes that root from the equation. You would then find only one answer instead of two. Factorising out x instead keeps both solutions safe.
To take square roots or to factorise easily, the coefficient of x squared should be 1. If it is not, divide both sides of the equation by that coefficient first. This keeps the equation balanced and ready to solve.