Lesson Guide
Explanation
Here is the key fact. If two numbers multiply to give zero, then at least one of them must be zero. This is called the null-factor law. So if x minus three, times x plus five, equals zero, then either x minus three is zero or x plus five is zero. The first gives x equals three. The second gives x equals negative five. A quadratic like this has two solutions, and you must give both.
Your Turn
Solve (x + 2)(x − 9) = 0. Check: x = −2 or x = 9.
Watch Out
x + 5 = 0 gives x = −5, not x = 5. Solve each bracket; do not just copy the number in it.
Explanation
Most quadratic equations are not factorised yet, so you factorise first. For x squared minus x minus twenty equals zero, find two numbers that multiply to negative twenty and add to negative one: negative five and four. That gives x minus five, times x plus four, equals zero, so x is five or x is negative four. Now check. The two roots add to one, which is the opposite of b, negative one. They multiply to negative twenty, which is c. Both checks pass.
Your Turn
Solve x2 + 8x + 15 = 0. Check: (x + 3)(x + 5) = 0, so x = −3 or x = −5; they add to −8 and multiply to 15.
Watch Out
The roots add to −b, not b. For x2 − x − 20 = 0 they add to +1.
Explanation
The null-factor law only works when one side is zero. For x squared plus three x equals twenty-eight, first subtract twenty-eight from both sides, to get x squared plus three x minus twenty-eight equals zero. Now factorise: seven and negative four multiply to negative twenty-eight and add to three. So x plus seven, times x minus four, equals zero, and x is negative seven or four. It is tempting to write x times x plus three equals twenty-eight and say x is twenty-eight. That is wrong: two numbers can multiply to twenty-eight in endless ways.
Your Turn
Solve x2 = 2x + 15. Check: x2 − 2x − 15 = 0, so (x − 5)(x + 3) = 0 and x = 5 or x = −3.
Watch Out
Forgetting to set = 0 first: x(x + 3) = 28 tells you nothing about each factor on its own.
Explanation
Look at x squared equals five x. It is very tempting to divide both sides by x and get x equals five. But that loses a solution. Zero works too: zero squared is zero, and five times zero is zero. Dividing by x is only allowed if x is not zero, so it throws that answer away. Instead, subtract five x to get x squared minus five x equals zero, take out the common factor x, and get x times x minus five equals zero. So x is zero or x is five.
Your Turn
Solve 3x2 = 12x. Check: 3x2 − 12x = 0, so 3x(x − 4) = 0 and x = 0 or x = 4.
Watch Out
Dividing by x loses the root x = 0. Always bring everything to one side and factorise.
Explanation
Every quadratic equation has a picture. The graph of y equals x squared minus two x minus eight is a parabola. It crosses the x-axis where y is zero, which is exactly where x squared minus two x minus eight equals zero. Factorise: x minus four, times x plus two. So the parabola crosses at x equals four and x equals negative two, and you can see both points on the graph. The roots add to two and multiply to negative eight, which matches the check. Quiz and exercises now.
Your Turn
Where does y = x2 − 6x + 5 cross the x-axis? Check: (x − 1)(x − 5) = 0, so at x = 1 and x = 5; they add to 6 and multiply to 5.
Watch Out
y = (x − 4)(x + 2) crosses at x = 4 and x = −2, not at x = −4 and x = 2. Set each bracket to 0.
📌 Key Points — this lesson in a nutshell
- Null-factor law: if a × b = 0, then a = 0 or b = 0. It only works when one side is 0.
- Method: rearrange to … = 0, factorise, set each factor to 0, solve each one.
- The root has the opposite sign to the number in the bracket: x + 3 = 0 gives x = −3.
- Never divide by x: x2 = 5x becomes x(x − 5) = 0, so x = 0 or x = 5. Dividing loses x = 0.
- Check: the roots of x2 + bx + c = 0 add to −b and multiply to c. They are the x-intercepts of y = x2 + bx + c.
Step 1: Introduction
Step 1 of 5
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