Graphing Quadratics Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each equation step by step to find the zeros (roots) of each quadratic function.

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First Equation: x² + 2x – 3 = 0

We need to factor this quadratic.

Look for two numbers that multiply to -3 and add to +2.

→ Those numbers are +3 and -1.

So, we can write:

(x + 3)(x – 1) = 0

Set each factor equal to zero:

x + 3 = 0 → x = -3
x – 1 = 0 → x = 1

✔ Two real roots: x = -3 and x = 1

This matches the graph — it crosses the x-axis twice.

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Second Equation: x² + 6x + 9 = 0

This looks like a perfect square trinomial.

Check: (x + 3)² = x² + 6x + 9 → Yes!

So:

(x + 3)² = 0

Take the square root of both sides:

x + 3 = 0 → x = -3

✔ One real root: x = -3 (a repeated root)

This matches the graph — it touches the x-axis at one point.

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Third Equation: (x – 1)² + 3 = 0

Subtract 3 from both sides:

(x – 1)² = -3

Now take the square root of both sides:

x – 1 = ±√(-3)

But √(-3) is not a real number — it’s imaginary.

So there are no real solutions.

✔ No real roots

This matches the graph — it never touches or crosses the x-axis.

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Final Answer:
For x² + 2x – 3 = 0: x = -3, x = 1
For x² + 6x + 9 = 0: x = -3
For (x – 1)² + 3 = 0: No real roots