EQN-03-07

Quadratic Equations

Solve x² problems by factoring or square-rooting — the two methods the GAT-E rewards.

~5 min read•Difficulty 3/5

Set equal to zero, then factor

x^{2} + 5 x + 6 = 0

(x + 2) (x + 3) = 0

x = - 2 or x = - 3

A product is zero only when one factor is zero — that "zero-product" rule gives both solutions.

Pure square-root type

If there is no middle term, isolate the square and take both roots.

x^{2} = 49 \to x = 7 or x = - 7 (don’ t forget the negative root)

Difference of squares

x^{2} - 16 = 0 \to (x - 4) (x + 4) = 0 \to x = 4 or x = - 4

How many solutions?

A quadratic usually has two solutions, sometimes one (a repeated root), sometimes none real. On the GAT-E the numbers are chosen so it factors cleanly — look for two integers that multiply to the constant and add to the middle coefficient.

Worked examples

Example 1 (difficulty 2/5)

Solve

: x^{2} - 7 x + 12 = 0.

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Solve

: 2 x^{2} = 50.

◎ Predict the answer first:

Tips

Move everything to one side $(= 0)$ before factoring.
For $x^{2} = k,$ write both roots $: + k$ and $- k .$
Look for two integers that multiply to the constant and add to the middle coefficient.

Common mistakes

Giving only the positive root of

x^{2} = k

Why: The positive root is the obvious one.

Fix: Squaring hides the sign — include the negative root too.

Factoring without first setting the equation to zero

Why: The zero-product rule is skipped.

Fix: The product-equals-zero step is what produces the solutions.

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