Factor quadratics to reveal zeros · lesson

Factoring x² + bx + c

💡 Find two numbers that multiply to c and add to b; they go in the two factors.

For x² + bx + c, look for two numbers that multiply to c and add to b.

If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs.

Check by multiplying the factors back out.

🧠 Let's work it out

Factor x² + x − 20.

What do you think comes next?

💡 A product is zero when a factor is zero, so each factor (x − r) gives the zero x = r.

A zero of f is an x that makes f(x) = 0.

If f(x) = (x − 6)(x + 2), it's zero when x − 6 = 0 or x + 2 = 0. So the zeros are 6 and −2.

Watch the signs: the factor (x + 2) gives the zero −2.

🧠 Let's work it out

Find the zeros of f(x) = (x − 6)(x + 2).

What do you think comes next?

💡 To find a quadratic's zeros, factor it first, then set each factor to zero.

If the quadratic isn't factored yet, factor it. Then use the zero product property.

The zeros come out with the opposite signs of the numbers in the factors. Check each zero by putting it back into the original quadratic.

🧠 Let's work it out

Find the zeros of x² − 3x − 10.

What do you think comes next?

💡 For ax² + bx + c, try factor pairs of a and of c until the outer and inner products add to b.

If a isn't 1, the factors look like (px + q)(rx + s), with p × r = a and q × s = c.

Try pairs, and check the middle term: the outer product plus the inner product must equal bx.

Guess and check is normal here. Multiply back out to confirm.

🧠 Let's work it out

Factor 3x² + 7x + 2.

What do you think comes next?