Quadratic Quest
Hunt down the roots of a quadratic: factorise it into two brackets, use the zero-product rule to read off the solutions, and see where those roots live on the parabola.
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Quadratic Quest
A quadratic contains an x² term, and solving one means finding its roots. The main weapon is factorising, from Expand & Conquer, plus one simple rule about zero:
Substitute it
Work out the value of x² + 5x + 6 when x = -2.
The zero-product rule
You just found that x = -2 makes x² + 5x + 6 equal zero. That is exactly what a root is, and here is the rule that lets you find them without guessing. If two things multiply to give zero, then at least one of them must be zero: If P × Q = 0, then P = 0 or Q = 0. Nothing else in arithmetic behaves like this. If P × Q = 12 you learn almost nothing about P; if P × Q = 0 you learn a great deal. That is why every method here ends with "= 0" on the right.
Mind the sign
One bracket of a factorised quadratic is (x + 2) = 0. What value of x does this give?
- x = -2
- x = 2
- x = -0.5
- x = 0
Solve by factorising
To solve x² + 5x + 6 = 0: 1. Factorise: two numbers multiply to 6 and add to 5, so 2 and 3, giving (x + 2)(x + 3) = 0. 2. Set each bracket to 0: x + 2 = 0 or x + 3 = 0. 3. Solve each: x = -2 or x = -3. Choosing the two numbers is easier once you read the signs first: - If the last number is positive, the two numbers have the same sign, and the middle number tells you which: both positive if it is positive, both negative if it is negative. - If the last number is negative, the two numbers have opposite signs.
Work it through
To solve x² + 7x + 10 = 0, look for two numbers that multiply to _____ and add to _____. They are 2 and 5, so the equation factorises to (x + 2)(x + 5) = 0. Setting each bracket to zero gives x = _____ and x = _____.
Solve it yourself
Solve x² - 7x + 12 = 0 by factorising. What is the SMALLER of the two roots?
When the last number is negative
In x² + x - 6 = 0 the last number is NEGATIVE. What does that tell you about the two numbers you are looking for?
- They have opposite signs: one positive and one negative
- They are both negative
- They are both positive
- One of them must be zero
When there is no middle term ◻️
One case looks alarming and is actually the easiest, and telling it apart from an ordinary quadratic takes one glance:
No middle term
Solve x² - 25 = 0. What is the POSITIVE root?
Match each quadratic to its roots
- x² + 5x + 6 = 0
- x² - 7x + 12 = 0
- x² + x - 6 = 0
- x² - 9 = 0
- x = -2 and -3
- x = 3 and 4
- x = 2 and -3
- x = 3 and -3
Roots on the graph
The roots you calculate are exactly where the parabola crosses the x-axis, because that is where y = 0. For y = (x - 2)(x - 4) the curve crosses at x = 2 and x = 4, so the crossing points are (2, 0) and (4, 0). Solving and graphing tell the same story two different ways, which is why a sketch is a useful check on your algebra.
Plot the root
The parabola y = (x - 2)(x - 4) crosses the x-axis at its roots. Plot the SMALLER root as a point on the x-axis (that is, the point (root, 0)).
The quest rules
A quadratic equation can have up to _____ roots. To solve it by factorising, write it as two brackets multiplied together equal to zero, then set each bracket equal to _____. If the last number is negative, the two numbers you are hunting for have _____ signs. On a graph, the roots are where the parabola crosses the _____-axis.