Does it make the left side 0?

The idea

A real number α is a root (solution) of ax² + bx + c = 0 if aα² + bα + c = 0; the roots of this equation are the zeroes of the polynomial ax² + bx + c.

Put 1 in place of x

Take 2x² − 3x + 1 = 0. Put x = 1: (2 × 1²) − (3 × 1) + 1 = 2 − 3 + 1 = 0. The left side became 0, the same as the right side.

Now try another number, x = 2. What happens to the left side?

• It is 0 again
• It is not 0
• It depends on the day

Test the numbers

Tap a number to put it in place of x. The working shows the value of the left side. Find every number that makes it 0.

Roots and zeroes

A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0. We also say x = α is a solution, or that α satisfies the equation.

Here 1 and 1/2 are the roots of 2x² − 3x + 1 = 0. Check 1/2: 2 × (1/2)² − 3 × (1/2) + 1 = 1/2 − 3/2 + 1 = 0.

Look at the graph: each root is a point where the curve y = 2x² − 3x + 1 meets the x-axis. So the roots of the equation are the zeroes of the polynomial ax² + bx + c. They are the same numbers.

Notes

α is a root of ax² + bx + c = 0 when aα² + bα + c = 0. The roots of the equation are the zeroes of the polynomial: where its graph meets the x-axis.

Check yourself

Is x = −1 a root of x² − 2x − 3 = 0?

What is the value of 6x² − x − 2 when x = 1/2?

Answer: -1

6 × 1/4 − 1/2 − 2 = 3/2 − 1/2 − 2 = −1. It is not 0, so 1/2 is not a root of 6x² − x − 2 = 0. (The roots are 2/3 and −1/2.)

For which value of k is x = 2 a root of x² − 5x + k = 0?

Answer: 6

4 − 10 + k = 0 gives k = 6. The equation x² − 5x + 6 = 0 has root 2 (and 3).

The roots of ax² + bx + c = 0 are the same numbers as…

• Yes: (−1)² − 2(−1) − 3 = 1 + 2 − 3 = 0 — correct. Yes! A root can be negative.
• No: a root cannot be negative. Roots can be negative. Put x = −1 in and see what the left side becomes.
• No: the left side becomes −4. Take care with signs: −2 × (−1) = +2, so the left side is 1 + 2 − 3 = 0.
• the zeroes of the polynomial ax² + bx + c — correct. Yes! A root makes the polynomial 0: it is a zero.
• the numbers a, b and c. a, b, c are the coefficients. The roots are the values of x that make the left side 0.
• the value of the polynomial at x = 0. The value at x = 0 is c. A root is an x for which the polynomial is 0.