Quadratic Equations · Grade 9

What is the quadratic formula and how do you use it?

The quadratic formula solves any quadratic equation from its three coefficients alone: minus the middle coefficient, plus or minus the square root of the middle coefficient squared minus four times the leading coefficient times the constant, all divided by twice the leading coefficient. It works even when the expression does not factorise.

Learning objectives

Identify the three coefficients

Before substituting, arrange the equation into standard form: a term in x squared, a term in x, a constant, all equal to zero. Without that arrangement the coefficients are not what they appear to be.

The sign is part of the coefficient. In an equation showing minus three x, the middle coefficient is minus three and not three, and this is the error that ruins more substitutions than any other.

A missing term means a coefficient of zero. If there is no x term, the middle coefficient is zero, and the formula handles it exactly as it handles any other number.

What the plus-or-minus does

The root is added once and subtracted once, which is how one formula produces two solutions. They sit symmetrically about the same centre, and the root determines how far apart they are.

If the expression under the root is zero, the two solutions merge into one. If it is negative, the root has no value and the equation has no real solution — that is the subject of the next lesson.

When not to use it

If the expression factorises at a glance, factorising is faster and safer. A difference of squares, a perfect square, or a trinomial with small numbers does not justify the formula.

The formula is the general tool for everything else, and especially where the solutions are not whole numbers. It never fails, but it does take more steps, and every step is a chance to slip in arithmetic.

Worked examples

  1. Solve with the formula: x squared minus 5x plus 6 equals zero

    1. Coefficients: leading 1, middle minus 5, constant 6
    2. Under the root: 25 minus 24, which is 1
    3. The root of 1 is 1
    4. Compute: 5 plus 1 over 2, and 5 minus 1 over 2

    Answer: x equals 3 or x equals 2

  2. Solve: 2x squared plus 3x minus 2 equals zero

    1. Coefficients: 2, 3 and minus 2
    2. Under the root: 9 minus four times 2 times minus 2, that is 9 plus 16
    3. That is 25, whose root is 5
    4. Compute: minus 3 plus 5 over 4, and minus 3 minus 5 over 4

    Answer: x equals one half, or x equals minus 2

  3. Solve: x squared minus 2x minus 4 equals zero

    1. Coefficients: 1, minus 2 and minus 4
    2. Under the root: 4 plus 16, that is 20
    3. 20 is not a perfect square, so keep the root
    4. Divide by 2 and simplify

    Answer: x equals 1 plus the root of 5, or 1 minus the root of 5

Common mistakes

Substituting a coefficient without its sign
Minus five x means a coefficient of minus five. Substituting five reverses the whole calculation, so write the three coefficients down at the side, signs included, before substituting.
Dividing only the root by twice the leading coefficient
The fraction bar runs under the whole numerator, including minus the middle coefficient. A partial division gives two numbers that are not solutions, and substituting back into the equation exposes it at once.
Using the formula on an equation that was not rearranged
The formula assumes the other side is zero. If a number is left on the other side, the constant coefficient is wrong and everything after it is wrong too.

What to remember

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