Polynomials & quadratics

Solve quadratics by factoring

ALEKS placement

Set the equation to zero, factor, then set each factor to zero.

What this covers

AB=0⇒A=0orB=0

Move every term to one side so the other side is 0. Zero times anything is zero, so a product is zero only when a factor is.
Factor completely: shared factor first, then a difference of squares or a trinomial pattern.
Set each factor equal to zero. A factor like 3x + 8 gives a fraction root, so solve that small equation fully.
Put each solution into the original equation. Its left and right sides must be equal.

Use factoring when you can recognize a common factor or a familiar quadratic pattern. Factors can give fractional solutions too. Completing the square and the quadratic formula are useful when factors are hard to find.

Worked example

Solve x² + 3x = 10

x2+3x−10=0The zero-product rule needs a zero on one side.
(x+5)(x−2)=0Five and negative two multiply to negative ten and add to three. They give the two factors.
x+5=0orx−2=0At least one factor must be zero for their product to be zero. Each equation is a separate possibility.
x=−5orx=2Solve each small equation. Either value makes the original left side equal ten.

Another worked example

Solve x² − 4x = 0 over the reals.

x(x−4)=0Factor out x; do not divide by x and silently discard x = 0.
x=0orx−4=0A product is zero when at least one factor is zero.
x∈{0,4}Both values satisfy the original equation.

A common mistake

A mistake Lemma catches

From x(x−4)=0

x−4=0x=0orx−4=0

“A product is 0 when either factor is 0, so x(x − 4) = 0 gives x = 0 or x − 4 = 0. Keeping only x − 4 = 0 — or dividing both sides by x — loses a solution: x = 0 works too, since 0 · (0 − 4) = 0.”

Try one

Sample problem

Solve the quadratic equation.

x2+7x−8=0

Practice solve by factoring free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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